While I agree with your overall premise, I'm not sure the "private school" piece is as big as some make it out to be. I began my career at a high school that currently costs over $70,000 a year to attend (as a boarding student.) We had kids who had almost nothing, and were there on Boys and Girls Club scholarships, or had been caught in war in Rwanda. We had other kids who were the children of rockstars and royalty (literally.)
You also have kids like Yianni and Spencer Lee. Both went to public school, but by all accounts, both are from families that are doing very very well for themselves. I'm sure there are plenty of others.