I'm not sure how I would define woman because it's so subjective.
No its not, its really not. And further, humans over millenia have never had any trouble figuring out who the women are when deciding who to oppress.
If I woke up tomorrow in a man's body I would still be a woman in my head. I'd feel all wrong like I was in the wrong body and wouldn't think of myself as a man.
Well, apart from the fact that this is entirely moot because it would never happen, I do agree that I would be pretty alarmed if I woke up tomorrow with a blokes body. But that would be entirely to do with the fact that I had spent almost 40 years with a female body, with breasts, a vagina, periods etc. So yes, I would feel pretty strange if suddenly I had a cock and balls!
I think being a woman extends beyond biology and to reduce it to biology is reductive to the experience of being a woman. I am not a woman because I have a female body, I am a woman because that is what lines up for me in my head.
And what about women who don't feel that way? Are they not women then?
What is the objective thing it that every woman and every transwoman has in common that no man shares? There isn't anything. There is no 'lady brain', no 'womanly essence' that we all share. Your female body makes you a woman, everything else is called 'personality' and there is a wide range of personality throughout womankind.
The only thing that every woman shares is the experiences of having a biololgically female body, whether it works 'properly' or not. And that is not something that transwomen share. And they might feel sad about that, but frankly, that is not women's problem to solve.
The idea of a 'lady brain' is incredibly regressive. That sort of concept is what was used as reason not to give women the vote, or certain jobs, or any sorts of rights throughout history. So slow hand clap to you for continuing to perpetuate it.