>The question is whether this is actually true cross culturally.

It's a fairly established finding in the field of human evolutionary psychology that they do. And numerous biological explanations for that have been articulated; the long and short of it is that women are more biologically invested in their sexuality than men are because they carry and give birth. Biologically speaking it carries more risk for them. Hence the behavior pattern whereby men initiate courtship.

So from a biology perspective it's fairly accepted that yes, much of the generalities you speak of are indeed cross-cultural phenomena. Though the specifics of sexual norms may vary widely by culture.