Earlier quoted context omitted.
"Private schools could be outlawed. Rich parents who want their children to go to better schools can donate to their city's public education budget." This awful idea makes sense only if you think the state owns people's children. As a parent it is my right, within certain bounds, to decide how my children are educated.
The state already imposes certain basic aspects of welfare on children, from vaccines to the fact that they have to go to school at all. In fact (I believe) private schools already have to be certified by the state. So it wouldn't be an enormous leap to simply stop certifying private organizations that want to play the role of state-certified educational institution.
If you ban physical after-schools, do you then ban online classes such as the math classes offered by the Art of Problem Solving?