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)
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.
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.