There are a lot of valid complaints about the way Stack Overflow moderates in general, but I don't get the impression that this particular squabble is over moderation practices -- it seems to be about upcoming changes to SO's Code of Conduct. I don't know what those changes are, so I'll reserve judgement. I am certainly hoping that it doesn't boil down to "I cannot abide being required to refer to transgender women with 'her' and 'she'," though.
If this is true, it's damning:
> Now if I avoid pronouns altogether by sticking to proper names or disengaging from the individual, that's being considered an insult too.
The complaints seem to be about some proposed new rule requiring users to explicitly write other users' pronouns, for no other reason than as a performative display of respect. This is quite different from a rule that says "If a user states that they prefer pronoun X, don't use other pronouns."
If moderators are really, truly being directed the way that quote suggests, then yeah, that's a bad directive, whether or not the intentions are good. But some comments here suggested that this had to do with a trans moderator, not a trans user, which makes me suspect the pronouns in question may have been not on public-facing Stack Exchange but in moderator chats/forums. And if moderator X is pointedly avoiding ever "engaging" with moderator Y when the position requires it, then that's going to be a personnel problem.
Those imposing the CoC are getting exactly what they want - driving anyone who disagrees from the field.