The sad thing about this issue is that it has become impossible to have a reasoned discussion about it, as people such as Kathleen Stock and J K Rowling have discovered.
Back in the '60s and '70s, the argument was that gender was a social construct, the role that society imposed on people according to their sex. Hence the campaign against ideas such as "girls wear pink", "boys play with guns", "real men don't change nappies", "women aren't logical", and so on. I won't say that the battle was won but certainly there was some significant progress. (Although the success which companies have had in selling the idea that "all little girls want to be princesses" makes me wonder!) Now, with trans activism being so mainstream, we seem to have slid back to a 1950s idea of womanhood. I have never seen a description of trans identity which didn't describe a little boy who wants to wear dresses or a man who longs for make-up and high heels. Conversely, transmen's girlhoods always seem to cite a lack of interest in dolls and a great many memories of rough-and-tumble games with boys. The trans identity is always built around stereotypes.
I think this is understandable. People say things like "I always knew I was a girl" or "I knew I was in the wrong body." How do you know? For instance, what does being a woman feel like? I am biologically female, been through girlhood, female puberty, and adult female experiences such as pregnancy, childbirth and menopause, and I couldn't tell you what it feels like to be a woman. In fact, if you rule out biology, how would you know that you're a woman? I think you'd only know by the way society treats you. If you were treated as a man, and your biology was ignored, you'd probably feel that you were a man.
The question that I am trying to ask is whether the supposed inner awareness of "really" being a man or a woman actually exists. I'm not convinced that it does. It seems more likely that some people feel that they don't fit the gender role of their sex, and that they want to change it. That's understandable too. You might feel that you would be more comfortable if you behaved and were treated according to the gender role traditionally ascribed to the opposite sex. Hence the re-emergence of stereotypes - it's easier to say "I want to act like, and be treated like, a woman" if you're wearing long hair, a pretty dress and high heels. Or, in the case of transmen, the requisite stereotypes.
Why is this unacceptable? Why do it have to be some inner female or male self that is trapped in the wrong body? (Some people say that, for instance, transwomen have "female brains". Female brains? What is this, the 19th century?") Why do transwomen have to be categorised as the same of biological women? Why can't they be accepted as transwomen, ie men who are comfortable in the female gender role? This would mean that many issues surrounding women's rights would cease to be problematic, if it was accepted that there was a profound difference between biological women and transwomen, and the latter should be accepted as having a different but respected identity.
If you don't accept that "transwomen are women", you supposedly hate trans people and don't believe that they should exist. This is ridiculous. There are undoubtedly bigots who won't accept transpeople at all, but there must be very many who believe that everyone has the right to choose their own lifestyle (as long as it doesn't hurt anyone) but don't believe that biological sex is unimportant.