I think private school is a bit of insurance - expensive at that, some kids will succeed regardless, but it might just give a slight edge for some to get into the course of their choice.
I don't buy your argument that private school is to limit based on religious, cultural or social grounds: most parents just want to give the best opportunity to their kids and see this as a good way to do so.
If private school teachers are better remunerated (more capable of attracting teaching talent) and have better facilities, QED the outcomes will be better.
