But I am highlighting some serious problems with the idea that it is completely or mostly innate.
Some of it may be, but individual professions change. In the medical field for instance, men preferred to have positions of authority which established, again, professional prestige. The problem was when it came to, say, women's health, they weren't all that knowledgable. Midwives and other trusted women used methods not embraced by the professional men doctors. Later it turned out that they had started to embrace women's medical knowledge and attempted to usurp the traditionally-female dominated practices. Once again, they used their prestige and their scholarship as justifications to remove women from their place in society. Nurses, for instance, became the woman's domain and were subservient to the trained male in medicine.
As I said, what became "men's work" and "women's work" shifted over time. However, a somewhat consistent rhetorical basis remained to justify those changes. Women were the nurturers, but they weren't that smart. Men on the other hand, were professionals, smart, and public figures, but not nurturers.