… on a social level, the characteristics we associate with each gender do not stem from biological, anatomical differences but from cultural norms.

What if this were false? Would it change your views? Does it not make sense to you that the larger sex should specialize in defending the smaller, child-carrying, lactating sex while the latter specializes in making a home for her babies and their father? Are tigers “socially conditioned” to hunt? Should they try kale?

I am not here to defend any of the circumstantially contingent consequences of biological differences. Lots of bad things “stem from” our “natural” state; just ask Hobbes. Maybe gender has obsolesced as a social construct. But biological differences can have practical consequences to evolutionary strategy. So long as you make denying that fact of life a premise of your arguments, your conclusions will be unreliable.