Why has the nutty fundamentalist strain of this belief got so embedded
Because trans ideology has become a cult and is executing a purity spiral
It certainly started out more reasonable. The original demands were for us to use preferred names and pronouns for people with gender dysphoria who had had medical modifications, whilst acknowledging that that people couldn't actually change sex. It was a polite fiction. Many of us were prepared to go along with that.
Then there was a gradual shift towards extending the 'trans umbrella' to cover various other groups who had no intention of undergoing medical modifications, some of whom didn't even want to be the opposite sex. And the concurrent blurring of sex and gender. Eventually, gender identity superseded sex for determining whether people are men or women .
Once gender identity supersedes sex, you have an irrational ideology. And it ends up executing a purity spiral as its believers try to outdo each other in zealotry.
It's had terrible repurcussions for various groups of people. Ironically, it's also made it harder for the very same group of people it was originally meant to help - people with genuine gender dysphoria who have undergone medical modifications but who don't necessarily see themselves as having actually changed sex. I think @Seethlaw is part of this group (apologies if this isn't the case and I've misinterpreted your position)