Your second question is tricky to answer, so I am going to ignore it. :D
For your first question, I always think the key is choice and support for people who make choices that are counter to the norm. Sure, it might be that when given a choice free of social pressures, more women might still choose to be the primary caregiver for children than men will.
That is fine, and women and men who make that socially traditional choice should be supported and not judged. The key, however, is to make sure we also support and don't judge the people who choose other arrangements for child caregiving. We should not design laws and social conventions that punish those who choose to go against the norm.
Part of my comment was questioning the meaning of support. Are we talking emotional support, and simply being accepting of other's identity? Are we taxing some roles and subsidizing others to normalize economic freedom? What if the market also has inherent preferences that mirror natural preferences? Do we actively recruit youth and try to get them into under-represented roles? What about undesirable roles?
I don't know... maybe this is all moot if we're still working through the "simpler" stuff like balancing laws and getting rid of stereotypes.
This is anecdata, but I'll tell you something from my experience. All the way until my studies, I met one girl who was interested in programming (more in general tech, but let's limit this to programming). During 5 years of studies I met the next 4. That's compared to probably ~200 guys. That was only around a decade ago.
Nowadays, I see tens of young girls joining local raspberry jams for fun in the UK. My friend's daughters are playing with electrical circuits and installing switch&lamp circuit between beds to chat in Morse code. And none of this comes from pushing for forced equality-in-numbers. This is just kids getting excited about stuff, because nobody is stopping them.
I think the stereotypes played a big part for a long time and we'll see a slow rebalancing of candidates' genders. I think (and hope) the situation will be very much different in ~15 years when current kids finish universities. But I don't believe much active support can change anything for near-future graduates.
So.. either parent who care gives can get the same leave for that caregiving. Basically, do not write gender specific laws. Write laws based on parenting role, and it can apply to either gender who takes on that role.
I don't think we can try to fix the market demand for the various roles, but we can certainly fix the legal discrepancies.
The emotional support is impossible to legislate, but I think as a society we should not judge negatively men who decide to be the primary child care provider nor judge women who decide to be the breadwinner (although in most families, both parents need to be both... I am a big proponent of day care)
In the case of the software industry in the US, the companies themselves have a strong economic incentive to get school girls to choose engineering careers. It'd almost double the pool of hiring candidates. So it's something that the market eventually balances, only maybe not fast enough.
That's this part of your comment: "emotional support", "accepting of other's identity", "taxing some roles and subsidizing others", "undesirable roles". Those all come out of stereotypes.
For your first question, I always think the key is choice and support for people who make choices that are counter to the norm. Sure, it might be that when given a choice free of social pressures, more women might still choose to be the primary caregiver for children than men will.
That is fine, and women and men who make that socially traditional choice should be supported and not judged. The key, however, is to make sure we also support and don't judge the people who choose other arrangements for child caregiving. We should not design laws and social conventions that punish those who choose to go against the norm.