I don't think we can reject the possibility that functionally there are some differences between male and female cognition. Especially once you include hormones. It's dangerous, IMO, to hang to much on the rejection of any differences of that type between men and women.
Again, you're right and again that's not what proponents of the doctrine of gender identity claim.
Like you, I believe that there may be found to be some differences between men and women besides the obvious physiological ones, and I think the most promising research may be around aggression, its presence and the lack of it. We're not there yet, but I expect that we'll find some few differences are innate (on average). Many other differences have been found to be due to nature rather than nurture of course, and given that our brains are both plastic and elastic, there's typically no way to say whether an observed difference in the brains of male and female adults isn't due to nurture rather than nature.
Again however that is not what proponents of the doctrine of gender identity are concerned with. They claim that preferences for traits, behaviours and mannerisms associated with femininity or masculinity are innate. You are born with them and they don't develop, they emerge like a butterfly from a cocoon and the child merely becomes aware of them.
That's clearly bollocks. As is the often found idea that a man becomes unable to do certain things once he identifies as a woman, because those are not womanly things.
Ladybrain = unable to do DIY, to read a map, to assert oneself, likes to watch chickflicks and cry, wear makeup and high heels etc etc basically it's a neat package of the most sexist stereotypes bundled up and projected onto the concept of male and female brains.
And there is no evidence for that kind of innate binary in neurological research. There cannot be since sex stereotypes and sex role stereotypes vary between places and times.