(That ‘drive’ in each case may be biological or societal or - most likely - a combination. It is only valid to see it as ‘prejudice’ ir ‘wrong’ if the person is told they can’t, or shouldn’t, follow the opposite path because of their gender, or if the person is discriminated against in their chosen path because of their gender)
But often the 'telling' isn't that explicit. And neither is the discrimination. If it were so overt that would make life so much easier!
It's a bit more complicated than that. You've already mentioned how society /work places view women and childcare....this is a particular issue.
If childcare is viewed as the responsibility of women then this will impact a whole range of things. There is evidence of it influencing career decisions at a young age( girls will favour careers which are viewed as flexible in preparation for having a family in a way that boys/men just don't). These careers often pay less.
Women are more likely to work part time which impacts earnings and career progression. Men either don't ask for flexible working or are more likely to have it refused.
The way society views men who take shared parental leave or become a SAHP is less than favourable. This is in part due to societal views on the role of men as financial providers.