Ya but black people are a real thing.
Transsexuality is also a real thing. But yes, it's not us actual transsexuals who should be the main reason against segregated bath- and locker rooms. We are only the tip of the iceberg. It's rather those much more common people who aren't really transsexual (i.e. have no disphoria), but nevertheless don't identify with the role society gives them based on their genitals who show that sexuality and gender isn't something that can be fit into 2 neat boxes.
In other words, why segregate people based on their sex or gender when they want to use a toilet or change clothes, if you can't reasonably segregate them in any other aspect of life?
Anyone can let their hair grow long and wear lipstick and a dress. But if he still has male anatomy between his legs he is a male no matter what he calls himself. Michael Jackson died with light skin and straight hair but he was still Afro-American. Some things cannot be changed just by calling yourself something you are not.
There seems to be some confusion between the different aspects of sex and gender.
A transwoman is a male regarding certain aspects of her genetics and anatomy, i.e. her biological sex.
She can have treatments and surgeries that change her anatomy, thereby making her anatomy more closely resemble a female. It's a matter of definition what biological sex a person has if they had a complete sexual reassignment surgery.
A transwoman is a female regarding her psyche, i.e. her gender.
Due to her upbringing she may have certain aspects of male gender roles internalized, but that depends on her personality.
Her gender however doesn't depend on the anatomy of the body she has.