Good humor, but “Stop paying teachers a living wage.” is actually a part of the solution. Vouchers that you describe that follow families is another part.
Halakha allows unrestricted competition between teachers, leading to lower prices and higher affordability. Let teachers or small group of teachers open one-class schools, teaching second grade only in their home or any small office or inside a school building to which they pay rent. R Kamenetsky and R Ruderman were in such classes in their Litvishe shtetl and grew up OK. They did complain about one insensitive second grade teacher, but our current towns should provide enough competition.