poppy
I don't believe it's so that they can abuse women/girls
I don't think it's necessarily abuse in the way that you are thinking of it.
Autogynephilia is a cross-dressing fetish. The person who has it is aroused by the idea of themselves as a woman. Hence those who go for hyper sexualised clothing and get a kinky kick out of all the female accoutrements. Because it is sex based and because it's the biology of women they want to possess, being in women's private spaces, particularly, is the goal. More than say joining a women's knitting circle.
It's very bizarre in that they actually want to become the object of their own desire. Almost more than desiring other women. (A common theme on trans threads is imagining themselves being penetrated as a woman, even whilst remaining attracted to women. Hence male lesbian).
So yes, having access to women's personal space is a massive turn on. It's not so much that they will jump on you (any more than any other man), but they will be getting off on you using lipstick, operating a Tampax machine, urinating. All that.
Separate toilets are useless for that kind of fetish. And every time someone suggests at school, in institutions and shops the transactivists go mad. It's partly validation, yes but it's also because of the kink.
Frankly if a man is demanding access to women and children, and is not content with just a place to pee, my first question would be why? The very insistence should raise red flags everywhere.
They never campaign for their own rights, their own spaces. Only those of women.