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.