I wanted to jump back to an older message:

This sounds vaguely like the "voucher" proposals that have been bandied about. Public education, much like the various debates on national health insurance, is one of these things where everybody pays in (via property tax) and even those without kids in the system see a benefit (a better educated society). If/when you start allowing people to pull money out of the public pot and spent it privately, you're creating an odd scenario. You have public money going to less-regulated private schools. Arguably, public money should come with public constraints i.e., the very constraints that private schools often exist to work around. The biggest example constraint is that public schools don't recognize any one "true" religion to the expense of others. Private religious schools, pretty much by definition, would find such constraints unacceptable.

As a secondary issue, something like choose-your-own-school would be relatively effective in large cities, where there's enough demand for just about any possible kind of school. In smaller towns, though, there simply isn't enough student demand for schooling, which would mean that a bunch of competing private schools would never achieve the sort of critical mass that you need to have specialized teachers, making everybody poorer as a result.

(Aside: I was about to add "successful sports teams" as another side-effect of the lack of critical mass, except there are a number of sports-first private schools around the U.S. these days where other educational goals are unquestionably of secondary importance. Such parents will sometimes uproot themselves and move hundreds of miles to get their kid into one of these sports-focused schools. I'm confident that this isn't a good general-purpose solution to the school choice problem.)