One of the things a free-market education system would offer is competition on reputation - allowing customers to choose schools before ever going there based on short feedback loops of many individual customers attending and reporting. This is already done with the public-private college system we have now - and yet you argue it's not possible to determine the quality of a school without going there, finding out it's bad and then being damaged?

If you don't understand how markets better allocate resources than top down rationing systems I suggest you go take economics at the nearest community college.

One of the concepts in the micro-economics principles class you will learn is that education has positive external benefits (beyond the market). In other words there is a marginal social benefit to education that exceeds the market demand. The best way to address this is not to have public school system monopolies of supply and take the cost to near zero - a much more efficient mechanism is to harness (superior) market resource allocation through demand subsidies.