Well, the highest maternal workforce participation rates are found in the Nordic countries and the Faroe Isles. I believe child MH is considered much better than the international average in these places. Of course, it's normal to get about a year off work on parental leave in these countries, so very young babies are not usually in nurseries, but then that's not the norm in the UK either.
The worst child MH is found in countries like South Korea, where mothers are pressured to give up work and spend their lives frantically organizing stuff for their kids. I don't mean that the SAH thing is actually making their children's MH bad, I just think the two things have nothing to do with each other.
Screentime and phone usage are what we should be looking at if we are worried about child MH. And frankly, if we pay parents to be at home, uptake will be highest among the least educated groups of parents, who are the least likely to be giving their children the wholesome outdoor play SAH experience. A lot of those kids will be getting a lot of TV while mum scrolls her phone.