I thought about this more overnight. This is where I've got to:
Transgenderism is contradictory and contains both ideas at once: a) there is no difference between the sexes hence I can be whatever I want and b) the difference is the most important thing in the world.
I think that which one you focus on may depend on other things about how you see the world.
To me, it always seemed that transgenderism hinders the process of reducing gendered socialisation, as it says that if you are gender non-conforming you must be the opposite sex.
However, I can also see the opposite position, particularly as regarding the abolition of single sex spaces. If you make all spaces mixed sex, as some of this is leading towards (mixed sex toilets etc), it reduces the tendency for people to segregate by sex, and thus is likely to reduce the behavioural differences.
I think that a lot of feminists view it like this: while there are still, alas, gender roles in our society, enforced by the bad people, some people will find their role intolerable, so it makes sense that we allow them to switch sex class. And it makes no real difference anyway, as there are no real differences between the sexes, so it doesn't matter how we classify people.
I think that blank slate feminism would struggle to find a justification for TWAW. But it also leads to - it makes zero difference whether they are or not. If you are a practical person who isn't that interested in theory, its as good as true, right? If there is no reason to care.
So basically, I do think that Biggs has a point. But I also feel in my gut that this isn't the full story. I think that there is more to the female support for gender ID ideology than this, and I'm not sure what it is.