He seems happy to keep the women in their place though.
Not this weak-ass argument again. With Uhura taking a far larger role in these films (compared to a secretary in space in TOS and the majority of the first six films) this argument holds no grounds. Especially with STiD where she is given more action. She is shown to be highly intelligent, motivated, strong willed, and most of all, more than just a "sex symbol" in Trek.
So really I want to add to what you said, yes he's happy to keep the women in their place, right alongside the men and many times, leading them.
And thank God.