No. Many transsexual people don’t want to be recognised as trans anything. They want to live and be recognised as the men and women the feel they are and have transitioned to.
And therein lies the issues.
Trans people transition to the gender they identify themselves as but are materially not. And there are so many genders these days.
A transitioned male transitions to what their perception of a female and their perception only. They do not transition to the material reality of a female’s lived experience based on their life since birth.
And what about a transitioned female who is non-binary or one of the other very many genders? What rights do they need?
In the case of the female, they still need sex-based rights. Even if they are ‘cat gender’ they still need sex based rights.
There are conflicts and it is ridiculous to deny there is conflicts. And there is also this confusion where the ‘woman/man’ genders seem to get all the priority. But when you start to unravel it, it is far better that trans people have their own rights that reflect their needs, but they retain sex based rights if they are female. Because if you take this argument that ‘transitioned males should have access to female sex based rights’ where does that leave the non-sexed genders? Does that create a hierarchy there that definitely should not be allowed to exist?
Transitioned males should have access to trans rights but not female based sex rights and protections of having single sex spaces, the right of having equal and sex specific representation, the opportunities set up to encourage female achievement after centuries of negative sexist discrimination (that still is around in abundance), in sport, education, employment etc.
Those sex based rights and protections came into being because of females having a body formed around one reproductive function whether that function is/has/can ever happening/happened/happen.