No one is "segregated by sex";
Of course they are - or were.
if we're talking about social identity, we're having a conversation about gender. And if we're saying "people of this category get to use this special room, but only people in that category get to do this ritual together", we're talking about social identity. How we choose to define, respond to, and exert political power over perceptions of sex is a purely cultural question, which different societies can and have devised different solutions to. Drag shows have only existed for two hundred years, they are not connected to the scientific discussion of biological sex traits except obliquely as part of our wider cultural conversation about gender.
And I do agree that transwomen are biologically female, at least in some respects. Your mind is no less a part of your body than any other part of you.
By 'mind' I assume you mean 'brain', and no. Having certain thoughts in your head does not change your sex or cause you to be the opposite sex.
So you are still ignoring studies which have identified structures in the brain that more closely resemble the sex the trans person is claiming than the sex they were assigned at birth?
What about them? Gay men have some brain structures that more closely resemble those of straight women than they do other men. Gay men are not women.
You still seem to be struggling over personal identity... and the basis of it.
My experience based on two people that I have known that suffered traumatic brain trauma is that self-identity is up to the central nervous system, and utterly out of our self conscious control. Neither of the people thought they were the opposite sex after the injury. However, their personalities changed tremendously. My reserved grandmother became a love smitten teen while a person I went to school went from wild alpha male to quiet, simple, and reserved. The original versions of these people never returned. So we have one body each and two personalities that didn't overlap.
So, if that can happen, and it has happened enough, the question becomes, why is it so hard to accept that there is a neurological construct that can develop for some people that make them think they are the opposite gender. We know there are some macroscopic differences between gender. We know there are psychological differences (when baselining) between gender. Why can't you make the jump to understanding that it is almost assuredly true that there are neurological portions of it to? You (and others) seem quite intent on wanting to draw a dividing line between genders, but when it comes to neurology, you suddenly become numb and disinterested in the science.
Regarding sexual attraction, we have people that have any number of fetishes... none of which anyone selected. How much of our preferences aren't remotely selected? We certainly don't choose our neuroses. We find things pleasing or unpleasing because our brain says we do. And that is all neurological. Chemical and physical reactions to stimuli we have no control over. Yet, for some reason, you think you get it all.