It's a good question.
First, my aim is not to ensure that women stay at home but only to recognize those who choose to do so. I have four daughters but they don't live at home now. One is a lawyer. She has a job and she leaves her two children in a childcare center for a few days. I have nothing against childcare centers.
However, if my daughter chose to stay at home, she would not get any financial benefits, even if she took care of her children's education on her own. I believe that the problem you raise about childcare centers is the one that we've raised also, and that is that children who use childcare centers receive some education services whereas those who don't use them don't receive those services. Personally, I have taken care of my children's education at home. I have used the same programs, which is not your case: you give them to one group but not to the other.