Just how could those in their ivory towers, their grammar school/private school/European finishing school/Russell Group uni/Cambridge Oxford elite educations, their widespread reading and travelling, their awareness of history and women's struggles and hard-fought rights, their knowledge of child development especially brain chemistry, and 101 other relevant things...how could they deny knowledge, natural justice, common sense, child safety...to pitch themselves to this particular wagon?
I think some of it stems from the notion (that started doing the rounds a few years before gender identity ideology in its current form took hold) that people are born gay and that's that.
As a lesbian, I did wonder at the time if that was definitely scientifically proven beyond all doubt, as I feel my instincts and lots of anecdotal evidence (including my own childhood and other factors that I strongly suspect might be relevant to my sexuality personally) suggest to me that while there might be a genetic susceptibility to homosexuality, I don't think it's that far-fetched to believe that very early childhood can influence sexual orientation too in some cases.
Furthermore, genetic predisposition and environmental factors aren't mutually exclusive when it comes to something as complex as psychology and sexual orientation, so I didn't see this as particularly controversial. Until I gingerly voiced this opinion to others in LGB circles and got lambasted in an almost identical way to the way gender critical people do nowadays. This was a few years before the gender identity stuff took off.
I was firmly told that "the science was settled" (I think it was something about penguins or someone's uncle or something) and that I was completely incorrect and terribly homophobic to question whether gay people are all definitely born gay or whether environmental factors might also play a part in some cases.
Despite thinking "well, I'm no psychologist but even I can see various potential patterns in the large number of gay people I know and their stories", I reconciled this personally on the basis that the "gay people are born gay" narrative was deemed necessary and very effective in the fight to end horrific gay conversion therapy practices. Obviously, because you can't change it if it's just how people come out of the womb, whereas if there's even an inkling that it might have been influenced by early childhood, parental attachment, etc, then it gives weight to the idea that perhaps it can be changed by conversion therapy. (To clarify, I don't believe conversion therapy works and I personally think that sexual orientation is probably fixed as whatever it ends up being by early childhood at the latest).
Also, it's not like anyone nowadays is going to study the causes of homosexuality, as what would be the point other than to argue for conversion therapy or to suggest that being gay is less desirable and/or problematic? I don't care that we don't know why people are gay, but it did bother me that people were adamant that we were definitely born this way, as it made me wonder where that left me if that's actually true - It made me wonder whether there were two subclasses of gay people and whether, if my homosexuality was rooted in my early childhood attachment issues as I believed, conversion therapy might actually work for me after all.
Incidentally, it's similar to how when non-trans people tell me they're definitely "cis" and that they feel their gender identity for themselves, it made me question whether there was something wrong with me if that's what all people, trans or otherwise are experiencing, because I don't feel mine.
I can't have been the only gay person who had this experience of being told to shut up and just support gay rights when questioning the science behind the mantra. So I learned that "gay people are born gay" was a helpful message in the fight for gay rights, even though I didn't take it to be literally true. i never really said it myself as I can't personally justify it and I think the science doesn't seem completely settled - I just stopped questioning it as it did seem to be a helpful narrative that advanced gay rights. I just saw it as a potentially useful shortcut to convince people that you can't choose not to be gay, no matter how hard you try.
When ideas such as TWAW came around, I initially viewed these in a similar way to the way I'd got on board with "gay people are born gay". I thought no-one meant it literally but that it was more of a idea, around treating people with respect, and suchlike. That's how most LGB people I know took it at first too.
Stonewall et al knew exactly how to sell a phrase like this as they'd already done it incredibly successfully with "some people are gay - get over it" (which is the same message as "gay people are born gay"). Plus, gay people like me were already fully on board with the notion that even if you don't believe a statement is literally true, it can still be a very helpful mantra in explaining a minority group's position to the general public who don't understand "us" at all.
I also think another factor is that gay rights have only come about relatively recently, and the times of misunderstanding, suspicion and horrific homophobic discrimination are so fresh in people's memories that it's very easy to draw false parallels with how gay people were treated very recently. It's still very raw and painful for many LGB people, and heterosexual people still experience a lot of collective guilt for all the prejudice, which often colours their judgment massively on trans issues.
Obviously these aren't the only factors in why intelligent people jumped on the gender train so quickly and willingly. But I do think that given that the LGB community and their existing ally base were used as vehicles to spread the trans message initially, the conditions were ripe for this to go exactly the way it has.