Your last paragraph: not quite.

First of all, the stereotypes of male and female psychology don't survive actual scientific testing, double-blind and peer-reviewed; women are just as aggressive, men are just as social and so on. Most of the stereotypes are indeed responses to societal expectation.

Except one.

Men in competitive situations will outperform themselves in noncompetitive ones. Women perform identically with or without competition.

And this is why "trans women" in women's sports is just wrong; even without the corporeal advantages of heavier male bones and muscles, that competitive male brain confers an unfair advantage.