Most private schools do better only because they weed out the expensive problems and force them back to the public system. The moment you randomize the students, the private schools drop back to the mean (or, generally, worse).

I find this unfortunate, because education is in dire need of some real, evidence-based, advances. We have a lot of new data about achievement and learning.

However, putting it to practice requires money, time and a LOT of effort. And you will have to fight the parents, too.