> For example, if stereotypical male/female attitudes and behaviors were found to have a substantial genetic component to them, I think a lot of socially progressive people would hate that.

I...disagree. I think you would find that most of the left takes as an article of faith (or, at least, a very strongly held belief) that gender identity and presentation (which necessarily includes stereotype-conforming as well as nonconforming identities and presentations) are very strongly genetically determined and that extreme distress results when people attempt (or are compelled) to mask what is genetically-determined in this area, and that that is fundamental to the belief that nonconforming identities and presentations should be accepted.

> There's a lot of emotional investment for many in the idea that stereotypically gendered behaviors are entirely or almost entirely caused by culture/social pressure.

There's a lot of emotional investment in the idea that the concept that there is a single acceptable relationship between external genitalia, gender identity, and gender presentation is a social construct, as well as in the necessary corollary that the particular relationship chosen in any particular culture that imposes such a single acceptable relationship is also a social construct. But that's intimately tied to the idea that each of those three things, while they may have some non-genetic contribution, is fairly strongly genetically directed, and while there may be some correlation between the genetic determinants of each, there is considerable individual variation, as well.