1/ You haven't mentioned how that SAHM must get a cooking credit, healthcare, retirement or house management credit or anything else in the litany of jobs required outside of immediate childcare and costs incurred by simply existing as a woman. Just a voucher for the hours, I assume, at which childcare would be open and none of the other hours
2/ A SAHP (thats stay at home parent) should be incentivized by raising wages and allowing life to be more affordable but your argument seems to be very focused on "moms" and "capitalist enterprises" and does not consider the reality that when SAHMs were more economically viable, it was not viable for all families.
Yes. And no. The gov gives the child X$ per week/month/year. The child parents use that money to take care of the child.
Society benefit from children that are well taken care of. Mechanisms to ensure that they are well taken care of are needed. Well funded daycare centers are one of the mechanism. A well funded household with a parent/grand parent/uncle is another one. In both case, an agency is in place to ensure the wellness level of the child.