If stats from dating websites are to be believed, all women tend to flock on all the same tiny percentage of men and ignore the rest. If women initiate, I’m afraid you will see only a few men getting any pairing. Men are likely to be less picky as we see in the wild.
Monogamy is half the reason this is stable on paper.
For initiating, men are less picky.
For marrying: I suspect they are about as picky?