CRUDE, as described in its description, is an incredibly helpful tool. There is nothing unhealthy about "feedback/discussion/requests of Close/Reopen/Undelete/Delete/Edit for questions and answers on Math SE". People are conflating what CRUDE actively is and what it is described to be, imho.
For instance, the close queue is not appropriate to the handling of controversial cases. "Extended discussions", for example, are not appropriate in comments. CRUDE can be a good place for that. The times I've used CRUDE, it was mostly with this intention in mind, as far as I remember.
More practically and close to the daily reality, the extreme inflow of poor-quality questions also makes the ability of swiftly dealing with them imperative, and CRUDE can be a tool for that. There can be objections with this: for example, the subjectivity of "poor-quality", or the circumvention of some core aspects of the close queue. However, I believe that it is "worth it". Again, that doesn't mean there is no objection, and we should hear those.
However, CRUDE, as is used, has a fair share of toxicity (imho). What I can't understand is why people think that this is "needed":
"Until the site stops being unhealthy there will be increasingly unhealthy responses to that."
"I agree with Hurkyl's diagnosis. The site became ill for having an unhealthy diet of PSQs and their answers for many a year. C.R.U.D.E. is the foul tasting medicine."
etc.
It is almost implied that in order to be effective, it has to be unhealthy. I don't understand this at all.
That said, I agree a hundred percent with @rschwieb here:
Now, when a problem and solutions exist, it is a simple matter to play the watchdog and complain about solutions you don't like. Having watchdogs is of limited value, though, when no serious competing solutions are apparent. (Given what's at stake here (digital content) I don't count "do nothing" as a competing solution.)
So, what are the "alternatives"? None. There is no need for one. We don't need to replace CRUDE. We just need to stop being toxic in an irrelevant way, and use it for its proposed topic: which is the discussion of questions and answers, as per its own description, and not users.
I believe what is "rude/polite" should be common sense. Apparently it is not, so I have the obligation to elaborate. Rules of thumb, which people can obviously disagree or not:
- Avoid sarcasm.
- Avoid adjectivizing users.
- If you are talking to X in a specific way because they are X, chances are you are being biased. So, if you disagree (or even dislike) X, do a mental check that your communication does not fundamentally rely on the fact that it is X that you are talking to.
- Assume good intentions.
Now, specifically to CRUDE, my suggestion is:
AVOID REFERRING TO USERS.
If you think some user is so disruptive that they warrant special treatment, flag them. Moderators are supposed to handle users. I don't think users should be able to, in effect, "moderate" other users directly. And I think the system agrees, since "targeted downvoting" is wrong, for example.
EDIT: I've seen arguments implying that adhering to politeness can conflict with accurately pointing out to users that they are wrong/that they are missing something etc. This is very far from true.
I present a few examples where I point that someone is incorrect or that someone points out that I am incorrect, and I don't see anyone perceiving even a glimpse of rudeness (note that they adhere to the "rules of thumb" I mentioned): [1], [2], [3], [4], [5], [6], [7], etc.
Furthermore, the interactions are rather short and straightforward.
Of course, as anything in this world, there are the exceptions. But this is what they are: exceptions. If you are not being rude, you will hardly have to handle resistance when you tell someone that they are wrong and properly justify it. $^{1}$ If you often face such resistance, maybe it is time to consider the possibility that yes, you are probably being rude.
There are cases, however, that interactions are not so simply a matter of correct/incorrect. For example, when they are related to perception (of MSE, of mathematics, of science etc). Even in those cases it is possible to offer criticism while avoiding being rude. Let me give some examples.
This is a case where I disagreed with how the user was using the website. A short discussion ensued, but I believe it was constructive and polite, and I would guess that the other user has the same impression.
This is a case where I thought the question was overwhelmed with information, and tried to point it out to OP.
Again, my suggestion is: when in doubt, follow the rules of thumb.
$^{1}$This is the impression I have with experience in this site and life itself, which is also corroborated by the links as evidence of such, at least w.r.t. the site.