Happy to have a debate in good faith. There are some issues with your premise, however. First off, plenty of women who are not transgender are adept at stereotypically male things, such as engineering and physics. Same goes for men, who may be wonderful at multi-tasking, great at nurturing professions, etc. The more we allow people to enter into fields not previously coded for them, the more we see that our notions about that being inherent to gender is just silly. Secondly, having the secondary sex characteristics of a gender is not always the definitive marker of who you are. Just because you have a penis doesn't mean you are necessarily a man. It is possible to have XXY chromosomes, some chromosomes that are XX and some that are XY, hormones levels that do not match your secondary sex characteristics, both a penis and ovaries, and any number of other conditions that do not adhere to a binary. Or, it is possible to have been born into the "wrong" body.
There may well be such a thing as a male brain and a female brain and I certainly would never presume to argue with someone about their own sense of identity. But what there is not is any evidence of different abilities, or even interests based on those differing brains. Those are all a function of culture. Part of how we know that is A) there is no reputable scientific evidence of difference B) different cultures and different time periods have different norms.
For example, prior to the 4th C BC, Egyptian women conducted most of the business, including legal business and men stayed home and did the weaving. Amongst the Mosuo of China, a matrilineal culture, there are functionally no husbands or fathers. Everyone lives in the house of their mother or grandmother and helps to raise the children of the women in that household. - who never live with or formally marry their partners. Grandfathers, uncles, and male siblings all help with childcare in a way that is uncommon in the West. Men build houses and engage in politics, women do the farming and most of the rest of the physical labor and they conduct business. Mothers are the most revered people in the culture.
There is a much greater difference between individuals than there is between men and women as a binary split.
Your idea that no-one can escape their culture is also just kind of absurd. There have always, always been people in every era and every culture who balked against their socialization and gone their own way. Quite often they paid a price for doing so, but it has always existed. In addition, culture would never change in any way if it was impossible to go against it. We’d all still be living in caves and wearing fur if we couldn’t ever change anything about our culture. I don't think you've really thought through that assertion because it doesn’t actually make sense.
The fact that transgender people exist doesn't disprove anything I've said. And, transgender people are also a tiny fraction of the population. Whereas the growing number of women in STEM, and the growing number of men in "caring" professions are not at all a small number. This is in spite of the gender indoctrination that begins at birth and is deeply ingrained in every aspect of our culture.
