Since the management Stack Exchange/Stack Overflow decided to keep silent on this, and we still want to understand and make sense of this, we are forced to analyse this as a black box problem (with perhaps a hole in it). This is how it looks from the outside:

1. To prevent that it can be used legally against them

   Nathan Oliver comments

   That could be used as an admission of guilt. I doubt anything that could be legally used against them is going to be put out into the public space anymore.

   This would be a very bad situation. You should not keep away a public apology and rectification, just to prevent potential legal charges. If the SE staf agrees that Monica has been such maltreated that a legal action is possible then also accept those possible charges.

   If one does not live up to it then maybe they do not feel guilty enough to accept the consequences?

2. Because they believe this will go away

   In Dutch we have a saying "wie geschoren wordt moet stil zitten", "if you are being shaved then sit still". Which means that when you are being scrutinized, you'd better not try to react to it (creating only more attention, cutting yourselves).

   Although that saying relates more specifically to a type of reaction that tries to defend or fight the scrutiny. It does not relate to reactions like apologies. Sitting still, trying to hide the evidence, if this is the case then the Streisand effect has made this a very bad strategy

3. Because they want to make a political point and abuse the Stack Exchange community for this purpose.

   This has been postulated in this question: Was Monica's firing intended to "disrupt the status quo"?

4. Because the top management of SE is acting like the stereotypical ‘Americanata‘ with an ‘Americentric’ view, pushing their ideas with lots of egocentric attention without caring for the other ideas (or not even thinking about it), and believes that many users on Stack Exchange are bigots and they do not care that we leave or are upset. They do not see the bad they do, or disregard it as not relevant.

   While this may not be true for the entire management this might be an effect and may be concluded from Twitter messages like this and David Fullerton mentions David Fullerton is "proud" of the current situation.

5. Because of an "us versus them mentality" and (some) staff of Stack Exchange has become hostile towards the content-contributors of Stack Exchange this might be explained by the staff having normalized criticism from the contributors and don't care about it or (better stated) try to avoid it. This is well explained in this answer from user568458

6. Because Stack Exchange believes Monica is wrong. This view may be due to a completely different look at pronouns.

   • On the one hand you may view the pronouns as words with a technical function, namely to substitute other pronouns and make speech and writing more easy and efficient. For instance, instead of "David Fullerton mentions David Fullerton is proud" it is easier to use "David Fullerton mentions he is proud" (although because of the compelled consistency one can not do that anymore if one wishes to avoid the confusing 'they' in, for example, "Bob mentions they are proud" for a Bob who prefers 'they/their')

   • On the other hand you may view that the pronouns, as a side-effect, also qualify the object/noun that they refer to / replace. This qualification may be number, 1st/2nd/3rd/4th point of view, number, social distance (tutoiement et le vouvoiement), or also gender.

     From this second viewpoint may stem the idea that one could have a right to a preferred pronoun. If a man can be called 'he' if a woman can be called 'she' then people who neither feel like man or woman should be entitled to a pronoun as well, whether it has this 'technical' function or not (be easy substitutes) does not matter. More specifically one is not only to be considered among the neutral clas 'they' but one should be even entitled to a very specific personally chosen pronoun, almost like a proper noun rather than a pronoun. Given that one believes in this right, then when this right is not granted/ignored it may be considered as if one's identity/person is not being acknowledged.

   SE is not giving us much information. We have to fill it in, and the only information we have does not look good (This story starts to live it's own life. Do something SE management!). Why join in bad tweets? Is it a mistake, then why no direct apology? Is this a fight against the bigotous transphobic Stack Exchange crowd (like in this corner of the internet)? In the more concessive way it is considered to be clear that at least 'Monica meant no harm' (but does that mean Monica's is still culpable for the harm?).

   The idea is strongly that using singular 'they' and neopronouns is not difficult at all and does not conflict with the view of pronouns as in the first place 'easy and efficient' replacers. There is no place for other views ("our CoC is not up for debate").

7. Because staff is following very strict orders from management.

   With this point I am imagining that the top management has worked out some sort of goal relating to inclusiveness and defined this in a very strict way. Now the people at the frontline have to solve it. But in practice it doesn't go so nice and possibly results in awkward measures. (I see this sometimes happen in bars and restaurants, where I see servers behave in certain unnatural and unpractical ways, but they have to because their bosses told them to not do it differently and not to ease off)

8. Because it is embarrassing

   First point 7 created a mess internally and on the outside. Second, now they do not want to admit it because of embarrassment. (especially difficult since the CEO is very fresh, CEO changes are not always so good for companies and this often has to do with culture)

I hope that it is not all of these eight reasons, because then we have a long way to go.

Isn't the simplest explanation because Stack Exchange believes the statements to be true? Or, at least, that they consider them accurate enough to not warrant a retraction?

I'm not saying I believe the claims are true (I honestly don't know), I'm saying it's likely Stack Exchange believes the claims are true.

1. The claims were made twice (in the original apology and in the Register article), and representatives haven't retracted it when they've posted, despite repeated extremely vocal requests.

2. From what I can tell, it seems that Monica's treatment goes against Stack Exchange's own (financial) self-interest. I find it hard to conceive of any plausible motivation behind Monica's demotion other than Stack Exchange actually believes what they say.

3. Conversations on the Internet are often interpreted differently by different people.

The statements might not have been retracted from the initial articles, but The Register has posted a new article that addresses — in my humble opinion — pretty much all of the issues raised here.

This new article — dated January 2, 2020 — was posted on The Register, “Stack Overflow makes peace with ousted moderator, wants to start New Year with 2020 vision on codes of conduct: Q&A biz admits mistakes, promises more discreet public communication.” Here is what it opens with:

“In a display of Yuletide good spirits, or possibly a desire to bury bad news, Stack Overflow has settled its beef with a former moderator and said she can apply to regain her moderator status.”

And here is an explanation as to why The Register explicitly named Monica in their original pieces:

“A Stack Overflow spokesperson subsequently confirmed Cellio was the individual referenced as ‘they’ in that statement and we reported as much.”

Much of the piece focuses on statements and actions that have clearly been addressed here since the article clearly links to items such as the December 23, 2019 apology posted by Sara Chipps. But this is also in the article:

“The Register asked Cellio whether she wished to comment further but she declined, citing the terms of her agreement with Stack Overflow. We also asked Stack Overflow whether Cellio had actually brought a lawsuit against the company, given that no such complaint is evident in the US court system's PACER database of filings.”

“A Stack Overflow spokesperson declined to answer. ‘We came to an agreement, and we can not provide any further details on the matter beyond what is in our statement,’ the spinner said.”

Additionally, while the initial statements made in the October 1, 2019 article in The Register still stand, that specific piece has been updated with this note:

“Cellio has now settled her case with Stack Overflow and has been told she may apply to regain her moderator designation. Cellio hasn't said whether she’ll do so.”