This is a subject I tend to avoid in general, as it is often fraught and runs the risk of causing unintentional offense, so I apologize in advance if anything in this response produces that effect. (I also haven't been following the issue, so it's possible this post has little value, except in regard to my own approach.)

I've personally made a practice of utilizing gender neutral language online for two important reasons:

1. Imperfect Information: This is a virtual environment and I don't know the gender identity of contributor. (I don't need to know b/c the function of SE is to assist people in getting answers to questions, which shouldn't involve identity.)

But this leads to the core rationale for adopting it

2. Optimal Strategy: Because I don't know, I utilize the strategy that is optimal. Gender neutral language casts the widest net--it won't offend people for whom it is important and shouldn't offend those for whom it is not.

My own views on gender identity are also irrelevant--all that matters is respecting people.

My goal as a mod and a contributor is to facilitate the greatest degree of engagement to continue to grow the Stacks and increase their utility.

Where I think it may become fraught is that, right or wrongly, this is divisive issue. To formally integrate mandated use in the COC, while it would provide reassurance to a segment of the community, would likely be viewed as divisive by another segment of the community.

I think Stack the company is in a difficult position of having to strike a middle-ground. (It is said in mediation that outcomes which produce lasting result is one in which both parties are equally satisfied or equally aggrieved.)

I don't know the answer, but the one thing I am confident of is that whatever approach is taken, it needs to be a minimax calculation--minimizing the maximum downside of a given action.

Gender neutral language is just good strategy, and not utilizing it diminishes utility.

But, sad as it is to say, being overt at a given time in pursuing this strategy is not necessarily optimal. It's a gnarly problem, with many dimensions.

So what exactly would be wrong with this policy for pronouns, anyways, rather than the current CoC?

If a user asks you not to refer to them as "he" or "she" (or not to refer to them as either), then you must stop doing it.

It's simple to understand, easily objectively enforceable, and covers the most common cases of misgendering. There's no list of neopronouns to memorize, no mind reading for the moderators, and no trolls insisting you identify them as attack helicopters. Just a simple rule that can be invoked with a simple request.

I have seen a lot of advocacy for the generic use of singular they, but I think that it does not work well in many circumstances. Calling it established, classy, or traditional is marketing hype not supported by facts: it lives in a small niche as a correlative to someone.

There are often stories about an individual and a group interacting with each other. It is really handy to have a plural they and a different singular pronoun to deal with this frequent scenario (note: any pronoun different from they does a good job here). With singular they there is a problem that the plural they is suddenly unusable.

Note that this is different from singular/plural you, because circumstances usually make sure whom you address, and it is also different from the use of singular we, for example, in a single-author scientific article.

EDIT: Some authors in the world of abstract games found it convenient to assign different pronouns to the abstract players of a game, see this question for examples and references.

EDIT2: If there is any neutral pronoun really growing in usage, that this is one. Compare these two Google ngram searches oneself vs. themself and five singular reflexive pronouns. I choose the reflexive because it is the only form that clearly distinguishes singular from plural. Note also the orders of magnitude: For 2000, himself and itself are of approx. same frequency, herself is one third of them, oneself one twentieth, and themself is one thirty-thousandth.

Does it work for all types of uses of pronouns?

It differs depending on whether or not there is an antecedent and what kind of antecedent.

Unspecific antecedent

I notice that, even as a non-native English speaking person I am already using singular their or them, quite often. So, certainly, there are some arguments to make for it and it may be no surprise that for some cases our brains might even process it faster. This happens especially in cases that are close to plural, or where there is not explicit person being mentioned.

Like when I ask google how to translate the Dutch sentence:

'elke persoon krijgt een eigen voornaamwoord' 

it becomes in English:

'every person gets their own pronoun'

(somehow the German "jeder Person bekommt ein eigenes Pronomen" get's translated with 'his' instead of 'they')

and it sounds close to the plural version

'alle personen krijgen een eigen voornaamwoord' 

becomes

'all persons get their own pronoun'

Such cases are not difficult to process to me. That might be because the subject 'every person', while being singular, also has a plural feel to it. It is also not ambiguous what 'their' refers to.

Specific and singular antecedent

More problematic would be 'the driver is sitting in their car' (is it the car of the driver mentioned in the same sentence, or the car owned by some group of people not mentioned in the sentence).

To me it does not always feel so natural to substitute 'he/she' with 'they'. And I believe I am not alone and also this is not just a problem for non-native English speakers.

For instance sentences like...

'Alice is moving their suitcases' 
'Bob has lactose intolerance and they do not drink milk.'
'Bob has lactose intolerance and they does not drink milk.'

... these sounds more strange with me. Especially the combination 'they + singular verb' sounds strange (While 'they', both singular and plural, should be conjugated with a plural verb, but this is not clear and when it is conjugated with a plural verb it is not good for clarity. It is not as common as 'you + plural verb'). Also, the use of they for a particular singular person is strange.

The pronoun 'they' works different from 'he' and 'she'

There have been many comments how the use of singular 'they/their' is correct and not abnormal in English. However, I believe that we need to differentiate in the use of the pronouns.

In that respect, in the quote that your take from the article that you refer to (where the processing of 'they' was found to be easier) you should highlight the particular case:

Firstly, when there was no explicit antecedent, a larger fronto-central positivity was observed 750 msec after pronoun onset for he/she than they, possibly reflecting the additional difficulty involved in establishing a referent for he/she than for they when no explicit referent is available.

The researchers in this article are in fact arguing that 'they' is not like 'he' and 'she'

Effectively, singular pronouns require an explicit antecedent, and require one immediately. The search procedure associated with plural pronouns, on the other hand, is less demanding of computational resources

In that research they did not investigate the use of 'they' in combination with a singular antecedent (they used a plural antecedent).

You might argue that the differences that they found between 'they' and the singular pronouns 'he/she' is exactly what makes the use of 'they' as not similar to 'he/she'. The singular 'they' might exist in English, but apparently we process it differently in our brains. Therefore: singular 'they' is not the same as singular 'he' and 'she'.

Based on my own troubles with such sentences (the Alice and Bob examples) I suspect that what makes singular/plural 'they' easy for sentences without antecedent, is exactly gonna make it difficult with (singular) antecedent.

Some side-note about one of the facts mentioned by the OP

• In 2015, The American Dialect Society chose they as the Word Of The Year for its ability to refer to all individuals as a collective rather than any specific gender-based word.

In the Netherlands 'genderneutraal' was voted for as most irritating word of the year 2017. (on the other hand 'regenboogtaal', or rainbowlanguage in English, was in the top 3 of word of the year 2017)

Now that I've had time to contemplate further, I'm presenting a more direct response. (My initial response has already received votes, so I didn't want to edit.)

From the FAQ on the COC:

Q9: Do I have to use pronouns I’m unfamiliar or uncomfortable with (e.g., neopronouns like xe, zir, ne... )?

Yes, if those are stated by the individual.

This is just a matter of good manners. "They" is not a form of gender mislabling because it specifically avoids gender—in using it we are addressing people as human beings. But, where a specific term is requested, if using pronouns, it is impolite not to respect the request, regardless of personal views or comfort.

Q11: If I’m uncomfortable with a particular pronoun, can I just avoid using it?

We are asking everyone to use all stated pronouns as you would naturally write. Explicitly avoiding using someone’s pronouns because you are uncomfortable is a way of refusing to recognize their identity and is a violation of the Code of Conduct.

Since pronouns have become an issue in the culture, my writing style in online forums has self-adapted to avoid them wherever possible. Scientific writing specifically strives for neutrality, here in the sense of dispassion.

However, if one goes to great lengths to avoid using a specific term, for instance a conspicuous number of mentions of a person's name to avoid using the requested pronoun, it becomes a form of protest, which can be fairly interpreted as a form of aggression.

Different people will have different responses—for some it might roll right off, where for others, it could be deeply troubling. If we follow a doctrine of kindness and respect, it becomes incumbent on us to consider the full range of responses, and pursue a course with the least chance of causing harm.

The problem I'm sensing is that some people have negative responses to neopronouns, and they feel that their viewpoint is not being considered.

But I think these are different. Where, in the latter case, it is a general discomfort with new terminology, in the former case it is deeply personal.

It is not the function of Stack the company, or Stack the community, to set the standards of the greater culture, but we do have a mandate of kindness and respect for the individual.

Thus we have an obligation to follow this mandate, regardless of our personal views.

(Kindness and charity do not imply comfort, and, in general, it is that which is most difficult that represents the highest expression of these qualities.)

• On the use of they

I'm not certain using of they in place of any pronoun is necessarily an aggression.

Specifically, my natural writing style has altered since pronouns became an issue, to avoid pronouns in general, and where using pronouns, avoid gendered forms. Even where use of they differs from a specific preferred pronoun, it is not inherently disrespectful. Because it is a natural form with longstanding historical usage, it may not be conspicuous in the way using a name repeatedly would be.

This, in and of itself, does not override the arguments I made previously because so much is dependent on context, but I think that Anathema's argument is sound and worthy of consideration.

It is possible that use of they as a universal default could be the true middle ground.

This, however, begs the question: "Am I using they universally, even in place of he or she, or only in place of neopronouns?"

This approach has many problems

Disparate impact against trans users

It's likely that some users in chat will refer to a female user as "she" and to another as "they", because they know the latter user is trans. This has already happened.

Despite being outwardly neutral, it is a form of misgendering: the speaker is going out of their way to show that they don't see the referent as their expressed gender, just like they would by saying "he".

The rules should not enshrine this behaviour as explicitly okay.

Hurts a small group of users

As you state, most non-binary people are happy to be referred to as "they". Some only accept "they", while others also accept other gender-neutral pronouns. However, some people find that "they" worsens their gender dysphoria, and require a specific pronoun. Some (7% in the linked survey) avoid pronouns entirely and prefer their name to be used every time.

"They" is still suitable in distant/generic references, as you propose, and of course in collective references, which are plural. But it is better for these users to use their pronoun or name in personal references: even if they logically agree with your argument, that won't help their dysphoria.

Not easier to navigate

Social norms are hard, fuzzy, and complicated. Using someone's stated pronouns is not: it's a purely algorithmic process. It could (maybe should?) be done automatically.

Unusual pronouns (but not "he" and "she") do increase overhead somewhat, but having more pronouns makes references less ambiguous.

"Neutrality" is ambiguous

There is a very good argument that the rules of Stack Exchange as a platform should not (appear to) take a side on a current hot-button political issue. Doing so risks destroying us as a place for technical work and turning us into a weapon in a political battle.

If Stack Overflow had existed in 1860, it would have been unwise for it to take an explicit anti-slavery stance, even though that stance was in fact clearly correct. (Also, the programming questions were terrible back then.)

However, not taking a stance is different from taking a neutral stance. Taking any stance on slavery would have been unwise; taking a neutral rather than opposed stance would have been evil.

The part where I explain why all the angry yelling

Recently, some people chanted "Faggots, you're going to hell" at a group I was walking with, because many of the group were visibly LBGTI. I don't enjoy that, but I get it: they think LGBTI people and their supporters are inferior and are going to hell. Nobody's confused about anyone's intentions.

In contrast, the posts here insist that they're moved by good will towards trans people. They come with lectures about how trans and non-binary users are looking to be offended, or making unreasonable demands, or ought to compromise, or are trying to take away a fundamental right to neutrality (this post). They're peppered with disclaimers about being respectful, polite, professional (this post), opposed to discrimination.

In a few cases, these claims are disingenuous. In most, they are perfectly sincere, but still unhelpful.

Given such good will, I would expect:

• Users who don't have the time to dig into the issue to simply stay away and wait for a consensus to emerge, rather than advocate neutrality.
• Users who do want to participate in the discussion to do a lot of research on why trans and non-binary people are so loud about pronouns, such as reading accounts of gender dysphoria. In particular, I'd expect evidence-based posts like this one to cite the literature on the tremendous effects of social support, even though it's not clear that the results generalise to an impersonal site.
• Openness (though of course not blind trust) to the idea that one's well-intentioned behaviour can be harmful. Instead, explaining those harms often feels like walking on eggshells, like some people are waiting for an excuse to say "Ha! You said I hate trans people, now I get to stop listening!".
• In particular, arguments that the current treatment of trans users is decent and not harmful, rather than blunt assertions.
• Acknowledgements of the needs of trans users, and attempts to balance them with other considerations (e.g. that paying attention to pronoun choice is an imposition, and that trans rights are a talking point for a specific left-wing ideology), rather than immediately treating the cons as dealbreakers.

Many users use analogies that aren't very much analogous, because they have not done this research. For example, one user is fond of "What if I find it disrespectful when people call me 'Ian' instead of 'Sir Ian Fripplebottom III Esq.'?". It sounds superficially similar to "I find it disrespectful when people call me 'they' instead of 'she'", but I've never met:

• Anyone who shyly asked a trusted friend to call them "Sir Ian Fripplebottom III Esq." and felt relief crash into them when they heard the name
• Anyone with a history of suicide attempts or alcoholism or self-harm that stopped when people stopped calling them "Ian"
• Anyone who fled a comfortable life for homelessness and dangerous jobs just so they could be known as "Sir Ian Fripplebottom III Esq."
• Any minor who committed suicide over their parents' refusal to call them "Sir Ian Fripplebottom III Esq.", and the grave still says "Ian"

People with enough trans friends to personally know the analogues of all these examples will find the analogy frankly insulting. It's obvious from their other behaviour that (typical) trans people desperately need to live as their gender, whereas (typical) people who get grumpy about titles don't have such pressing needs.

To put it in utilitarian terms, Sir Ian Fripplebottom III Esq. gets a very minor benefit from being called his full title; others suffer a similarly minor cost by saying that many syllables. He may be legitimately suspected of trying to impose his preference, not because it's much greater, but because he wants to show he's in control. "Respect is a two-way street" applies.

Trans people get immense benefits from being recognised socially as their gender. Even if the other party has better grievances than "but remembering pronouns is hard!", such as religious objections, the utilitarian calculus is massively in favour of the trans person.

Many users here don't seem to disagree with this utilitarian claim so much as ignore it.

As a result, people who write these posts believe they have shown good will and good faith and a willingness to compromise, whereas trans users and their supporters believe they've been met with a wall of disingenuous refusal to even notice their needs. No wonder both sides feel wronged!