You are either paying the mother (or father) to stay home, or paying for child care so they can work.

I don't think it has to be a completely level trade on an economic basis. A lot of parents would prefer to be able to stay home even if it's not a purely economic optimum. And staying at home doesn't necessarily preclude earning any income. A parent could still do part time work from home, or even outside the home, when the other parent is there etc.