There's really two components to gender identity: the socially constructed stuff, which is most of it, and the biological stuff. Usually changing the biological stuff just lends itself to better socially constructed stuff.

Most trans people I know, for example, have no desire to change their genitals. Probably, I'm guessing, because nobody sees that. So their genitals have pretty much no relation to their gender identity. Which makes sense when you think about it. I mean, I gender everyone in my life, including people I see only for a few seconds. But I see very, very little genitals. I'm really just guessing, and everyone is.

It's very complicated. Of course men can wear dressed and makeup and such, but that is a very high-friction activity, borderline dangerous. Women can do that, however. So then I question if there are transwomen out there who would be satisfied in a world where anyone can wear anything. Such a world does not exist, so we don't know.