Are you suggesting that the father should pay the mother a salary for raising the kids? (Or same thing with the sexes reversed if it's the father raising the kid)
Isn't that just a more formal arrangement of a stay-at-home mother/father? The main difference would be: more financial independence to the stay-at-home party, which is good in my book, but I'm not sure if that's what you're suggesting.
Of course some people prefer to raise their kids themselves, but that seems more like an expensive lifestyle choice than a necessity.
The economy of scale isn't as big as one might like. Child care staff ratios for infants and toddlers are something like 1:3 to 1:4. Between that and the cost of compliance, licensing, insurance, and the inevitable scandals when deployed at scale, (If you have 1 million caregivers, you absolutely will have some number of murders or molestations. Unfortunate when a parent does it to their own child, but A Public Scandal when a government employee does it) you have got to start wondering if it's worth the hassle.
Or you can have mothers take care of their own children. Which has been the default of every human society for tens of thousands of years.
Paying people to be stay at home parents would make it a viable choice for a lot more people. It would also help reverse the trend of the fertility rate dropping below 2.1
Well, until they run out anyway.