The issue is not public versus private schools.
Every study on the subject consistently concludes the same thing: that there is a proportionate relationship between socio-economics and educational performance. Private schools with rich kids do no better than Public schools with rich kids and conversely, poor kids in poor private schools (and yes they exist), Jim Bobs Christian Academy, etc. do just as badly as their commensurately poor public counterparts.
It is time to stop attacking the performance of public schools and instead, commit to making them better. Stop attacking public school teachers and start to thank them for doing jobs in schools where the requisite skills are actually higher. Teaching rich kids is easier; I've done both; believe me. It just so happens that public schools have become the repository for poor immigrants who are not even well educated in their native tongues, much less English. When we remove them from the statistics, public education is doing quite well.