I'm not seeing much of a push for women there, but more of a push of men out of there, or satisfying different criteria.

The behavioral stereotype that I'm seeing is that when a man considers a job that they might like, that has some socially acceptable status (teacher is considered a respectable profession compared to running a garbage truck, at least for an educated middle class family) but that pays lousy, then they more often than not discard that option as unacceptable and taboo and go looking for a job that sucks in other aspects but pays better e.g. driving a truck or building houses; women in the same situation more often than not stick around with lower pay.

If a stereotypical man needs to choose between money and reasonable hours that are compatible with seeing your family, they tend to choose money, the sterotypical woman often chooses the opposite (part of which is the argument in this article for the years after childbirth). A particular local example that I see is that the municipal public transport drivers are mostly women and long-range (both scheduled traffic and tourist trip) bus drivers around here are mostly men. The municipal transport pays much less, so men don't apply there; but the long-range drivers work obscene hours and are away from their families for most nights, so women (who have the needed experience qualifications since they're driving the same machine, just locally) don't apply there - there's a self selection.

In management, we can observe a pattern of increasing divorce rates (for the same age) as men reach higher levels of management; suggesting that there may be a pattern that when choosing between a possibility of promotions and not wrecking up your marriage, more men choose to prioritize their work and more women choose to prioritize their home life (which IMHO is the sane choice). The same applies for physical health - women tend to avoid many of the physical jobs that screw up your body or risk your life. E.g. roofing is a job that can be done well by women, but is one of the more lethal jobs in USA - and it has about 0.5% women in there. It's not a well paying job comparing to skilled jobs (e.g. a registered nurse) but it pays significantly more than childcare, but we're not seeing the poorly paid women in childcare joining up roofing just for the money.

Also, if a stereotypical middle class man needs to choose between doing a lower class job and sufficient money, they often choose sufficient money; the stereotypical middle class woman chooses otherwise. I have an observation from some time when our local economy was doing badly - if a man can't get a respectable job that can sustain their family, they'll often get a 'disrespectable' job below their skills or even occasionally commit suicide if they fail. On the other hand, women actually do overwhelmingly continue work in female-coded low-paying jobs like you describe even if other options exist; e.g. we had a local situation many years ago when schoolteachers were very, very poorly paid, but a construction boom (or bubble) had a big need for all kinds of workers. This resulted in almost all male teachers leaving schools and getting construction jobs, including very many that don't have a strength or skill requirement as e.g. painters, and very few female teachers did so (though I know some), resulting in the gender gap in school teaching becoming even more extreme. The same happened for university students picking their course subjects - because of the known problems with teacher pay, boys absolutely refused to study pedagogy degrees, with the gender ratio dropping from something like 30/70 to 1/99, but girls still enlisted. Another local observation is that boys treat the local med school entry conditions as all or nothing - if they can't get in to the doctor's "full medicine" study track, they refuse to go to med school at all and do something else; but girls who apply tend to choose both "full medicine" track and nursing as options, so if they want to be doctors but don't make the cut they consider the lower paying carreer path as acceptable. For career paths that pay poorly at the bottom and well at the top, young men will face an 'up or out' pressure from their families; if it doesn't seem that they'll reach the "good paying" level, then they'll be pressured to drop out of the career and do something menial but better paying; while for women its considered acceptable to stay there and hope to get supported by a spouse.

I could go on and on, but I've probably made my point - there seems to be a difference in preferences in job market. For men, decent pay compared to alternatives is a 'hygiene factor', and they'll sacrifice all kinds of other important job factors (hours, prestige, office vs outdoors, risk and health, family balance, abusive conditions) in order to avoid getting stuck in an otherwise decent but low-paying carreer. For a man, intentionally choosing the low-money path is essentially taboo, their family and society will shun them for that and push them towards various tracks where decent money can be made; but for a woman, it's not so, so they stay in low-paying areas that are otherwise rewarding.