Celestina, I think you may have hit on the nub of the issue!
You said,"I take your point about the parallels between white Britons and women, but I don't accept them. The first was/is the massively dominant and powerful group within society. Women aren't. . . . "
I agree that women aren't the most powerful and dominant group in society. However, some women ARE afforded more privileges than other women, or even other men who may be disadvantaged. And, I would argue that (and I'm braced for the slap,) women generally benefit from many privileges trans people don't enjoy, while trans people can be subject to many of the same disadvantages that women face (e.g. rape, sexual assault, sexual harassment, being sexually exploited, facing discrimination at work, etc.)
Feminists describe gender as a social construct and wish to see the abolition of the concepts of masculinity and femininity. I agree with this - these are subjective concepts in any case and used to exert conformity and control which specifically disadvantages women (not great for men either, but not quite in the same league.)
However, one might argue that the concept of patriarchal oppression - specifically that all women experience oppression in some form because they are female and all men enjoy the benefits of male privilege, whether they ask for this or not, is predicated on the existence of a gender binary!
So, if you introduce a group of people who don't fit neatly into the box of male or female, it upsets that apple cart, particularly where there is evidence that those that don't fit male or female are even more likely than women to suffer abuse, discrimination and exclusion because they don't fit anywhere within the gender binary - they are complete outlaws, transgressors, and therefore threatening to the identities of both men and women.
Is that perhaps where some of the hostility is coming from? Fear that recognition of trans people will undermine the whole concept of women being oppressed within patriarchy (if it's possible that another group may be even MORE systematically oppressed by it?)