What do you care what the traditional roles in Western society are, so long as you're both happy?

Presumably if women rarely initiate and instead expect men to approach them, a man who frequently approaches women will be much more likely to find sex/a relationship than a man who just waits around for women to do the initiating.

What fraction of the time do they succeed?

I don't know. Sometimes they do and sometimes they don't. But I am sure they succeed more than they would if, like me, they never tried.

(And when they do, how do you know that part of the reason why they had picked Susan rather than Jane in the first place was that on some level they already knew that they had less of a chance with the latter than with the former?)

That definitely is part of it.