Although I fault Dawkins greatly for his stance here, I think that most of the causal factors here can be traced back to the approaches that (some) (loud) trans rights advocates have taken.

Suppose you have a group that demands vociferously that they get to decide how to alter or restrict privileges held by a group in need of some protection, and when you challenge their rationale or hold up examples of how their positions could cause harm, their dominant response is making strident statements about how evil you are and how persecuted they are (despite having support from society that is outsized compared to their numbers). Furthermore, suppose you see them frequently badly distorting science in ways that, if true, would bolster their case (except it's not true). If you were Dawkins, what would you do?

When these were fundamentalist Christians, the answer was pretty clear: you respond with equal stridency, opposing their intolerance, belittling their misrepresentation of science, and goading them with difficult questions that provoke them into making fools of themselves.

The problem is that trans rights advocates have, as a group, managed to tick almost every single box there. I don't want to be unnecessarily antagonistic, so I'd rather not list examples, but if you wish I will try to do so in as respectful of a way as possible.

Dawkins knows how to play this game. He's been doing it for decades. It's ever so easy to play it again. It's much harder to ask whether he should be playing the same game or a different, trickier one, where one he recognizes that despite the manifestation, there's actually a core reality underneath that needs to be acknowledged.

The answer that the skeptical community is being divided by an "anti-trans moral panic" is too easy. I fault Dawkins heavily for not looking into the science more deeply (or not believing it), and for lending support too quickly to a document with anti-science provisions in it (or feeling that it was an acceptable compromise...though...I really doubt that; I expect he overlooked it, as the genuinely anti-science parts were pretty well hidden, as opposed to the offensive parts (which also sorta implied a rejection of science if you knew the science) which were on full display). But I also fault trans activists for being willing to tolerate activism that in many ways is almost indistinguishable from religious zealotry. There is an anti-skeptic trans advocacy fervor that has at least as much explanatory power for the schism as does anti-trans moral panic.

Anyway, it's wonderful that you are taking a thoughtful evidence-based approach (e.g. in your answer to Ritchford). But if you look at the response to, say, Dawkins, it's not all-evidence-all-the-time by trans advocates. That's the only way to shut down people with a Dawkins-like phenotype when they are on the wrong side of an issue. They weren't shamed or pressured or canceled into silence by the religious, even back when religion held much more sway than it does now; why would they be by anyone else?

(Evidence that this does work: I think the New Atheists had some a bit of the stuffing knocked out of them in their militancy against religion when faced by actual careful scientific work by people like Scott Atran and Jonathan Haidt, which really cut off some of their strongest attacks against religion--not that it was true but that the evils laid at its feet were in some cases laid unjustly. Even if some people (Sam Harris especially, though I'm not sure Hitchins lived long enough for us to tell about him) seemed never to accept the research, it at least allowed a reasonable number of listeners to move to a somewhat more nuanced view.)

It's really unfortunate, because trans people do face huge societal challenges, and there is very good evidence that there exists an underlying phenomenon that they are experiencing (as opposed to, say, some whim or inability to face reality, as some would like to believe). Skeptics are awesome at shooting down stupid reactionary emotional arguments when armed with good evidence. Skeptics ought to be the natural adversaries of the reactionary hateful anti-trans crowd.

I just don't see a robust path to get there with the style of trans advocacy that is most amplified by social media, because the advocacy contains prominent examples of practically everything the New Atheist types hate.