A lot of men can do "women jobs".
^How many women choose to
mend cars, go down sewers, mine etc.^
And for all their working lives?
Women were prevented from working down miles from 1842 to 1985, by which time there were far fewer mines, and just after the Miners' Strike.
Women have always worked, because some women had to, to avoid starvation. Things like marriage bars mostly affected the professions, but it created the expectation that women would give up work on marriage, even where there wasn't a formal marriage bar in place. This changed over time to expectations that women will give up work when they have their first child. There's still an expectation that most women will want to do that, or at least go part-time, but there isn't any such expectation that men will want to do that when they become fathers.
For most parents, it's an economic decision- women often do earn less than their husbands, because of being younger and less advanced in their careers, because of the gender pay gap and roles dominated by women being less valued, and because of equal pay, because it's still far too often the case that women do not get equal pay for equal work with their male peers despite it being half a century since that was law. Women get paid for maternity leave, more than men get for paternity leave, and for many couples, there's not really a choice. Even Sweden has financial incentives - Swedish colleague said they would get less money if he didn't take his paternity leave, even if his wife took more time. (I don't know how it works for single parents - what happens if you're widowed after becoming pregnant?)
So it seems to me that equally paid parental leave would make it a freer decision for most parents, if they won't lose out financially whichever parent takes leave to be the SAHP. This would mean that over time, more men would take more time out of careers, so it would become more socially acceptable and it would be harder to discriminate against women of childbearing age if you're just as likely to lose men to parenting for several months. Women would of course still have to take some time, but it's the long term absences employers seem not to like.
There is evidence that men are recruited and promoted for their potential, whereas women are more likely to need to prove experience - potential isn't enough. In STEM careers, women are very often more highly qualified than their male peers, for the same reason. And if women don't apply to jobs until they can show they match at least 80% of the requirements it's probably not just imposter syndrome, it's also because they will be judged more harshly and will need to show they match those requirements - recruiters are less likely to take a punt on them as they might with a less experienced/qualified man. If a woman stands up for herself in the workplace or asks for a parish, she is more likely to be seen as aggressive ot for it to be argued that she doesn't deserve it.
And it's bloody tiring having to fight that bit harder than you see your male colleagues doing, just to keep up. This is partly why the leaky pipeline happens - often it's something like maternity leave or redundancy forcing a woman's hand, a change in circumstances that means you have to review your career, and you end up thinking, fuck this, I'm taking time out, I'm changing my career to something where I'll be appreciated more, may not be paid more, but the hours are better, I can do fulfilling things outside of work, because I'm not going to get it in work.
I am childless, I am the only woman in my department, I have never been promoted, I have had to fight hard for payrises just to be equal, I am far more formally qualified than my peers. Many of the men work part time, but this might be to spend more time on their hobbies. They will use work as an excuse not to go to things like school nativities or sports days, as if no mother ever finds it as tedious as they do, but work is generally sufficiently flexible that they could go, but it's easier to lie about work than be honest that they don't always want to be a parent. They are heroes when they work from home with a sick child... but mostly, they still have more choice about when to be an active parent than the mothers of their children do.