Earlier quoted context omitted.
If the market were able to solve the problem better, private schools could just provide better service at lower costs (compared to public school cost + state subsidy) and few would still go to public schools right? State subsidies for public schools are not that large anymore (34% of budget for UC Berkeley comes from tuition+fees while 13% comes from state of California). These are actually IMO all examples of Baumol…
How are private schools supposed to offer better service at a lower cost, when public schools are effectively free? This is hardly a market. Even if people wish to move their children out of public schools en masse, it's not like governments are going to reduce property taxes accordingly.
It's actually even worse than that because you pay for public education even if you don't use it.
So in that regard, if you send your kids to private schools, you're paying for their education twice.
This makes it very difficult to compete, and as a rayiner pointed out, isn't a real market.
I think the best solution to this is vouchers, which is a nice blend of public dollars but competitive schools.