Consider the mixed messages straight men receive. Our sexuality is labeled "problematic", our interest is default assumed to be unwanted, badly timed or otherwise inappropriate, yet we still have to initiate. Moral panics have made the process difficult for decent men, since, excepting obvious forms of harassment, there's very little consensus.

Create a post on whether it's okay to ask a coworker out, or message a recent acquaintance at a hobby group, or a woman in our social circle, and they'll be a thousand different responses, a thousand different standards/opinions. And the cost of screwing up can be legitimately concerning. Being mislabeled a creep and/or creepy can destroy someone's social life.

Men who aren't bothered by this are usually the types who ignore boundaries anyway. Continuing the expectation for men to approach therefore selects for them.

On the other hand, women are rarely judged as a threat. Unreciprocated interest from women is considered benign. Sl*t shaming generally only applies to sexually aggressive approaches. The social cost they face amounts to not much more than embarrassment, which would change once this becomes normal.

Women approaching makes a lot more sense now. Outside of situations like bars/clubs, it's safer overall. They can get to know men first without being labelled devious Nice Girls.

They still get to select men who measure up in other areas of masculinity, just not this. This one thing - a man's courage in approaching - is no longer a factor in their dating choices. Seems like a minor loss.

If this is indeed just a social norm then changing it would benefit everyone.