The dilemma of education policy is that we'd like two things to be true about education:
1. Parents should be free to use their resources to help their children achieve academically.
2. Schools should enhance social justice by giving poor kids the education they need to become successful.
On one hand, we want young people to earn their own ways in life, so that they succeed or fail based on personal virtue rather than on family privilege. On the other hand, it would be an unnatural, totalitarian regime that would prevent well-heeled, well-educated parents from helping their kids learn more and achieve academically.
In my home town of Rock Island, Illinois, there's a "magnet" school. To get their kids into the school, parents agree to a certain minimum amount of involvement in their kids' work and the school's operation. Since parental involvement boosts student performance, kids at the magnet school do pretty well. Parents know that at the magnet school their kids are in classes with other kids whose parents are involved, kids who are going to do better academically, and who are less likely to slow the class down.
On one hand, the magnet school is great because it lets parents who are committed to their kids' educations segregate their kids, putting them with other kids whose parents are involved. The school encourages and rewards parental involvement. On the other hand, for the kid with the misfortune of having parents who won't or can't be involved, it's a loss. On average, the brighter kids get siphoned off from their neighborhood schools to the magnet school. Through no fault of their own, the kids left behind are now in classes with fewer bright kids, and their educations are going to suffer as a result.
Private schools are the same way, taking generally better students and parents out of the public school system. If rich parents had to send their kids to public schools, you'd see those schools improve, thanks to the influence of the kids and the efforts of the parents. But forcing parents to send their kids to public schools would hurt the parents' ability to provide better educations for their kids, so that's heavy-handed policy.
Government subsidies for private schooling would improve parents' ability to get their kids good educations, but they would also encourage those parents who take education the most seriously to take their kids out of public schools. That would leave the public schools with fewer committed parents and worse results for their kids.