I'm not trying to push people into roles they don't want (I think I've made clear that the approach I described is a simple one to start at, not a good policy). My model in this case (which doesn't really map to reality) is "gender preference for roles is unbiased uniform random".

I don't really think job preferences are that strongly encoded in gender, I think far more men could be kindergarten teachers and more women carpenters, although men who try to become teachers face a huge amount of extra work because people don't trust men with kids as much.