no, but increasingly the ruling class is being drawn from the ranks of the privately educated from almost birth. this trend will continue.
Not always true.
There are many small wealthy communities where everyone goes to public school, because the school is so well funded from local tax money and the community isn't large, the public school can be just as good as an elite private school.
There's a big difference between attending a public school in an affluent suburb and a private school (e.g. Groton). In the first case, most students will go to college, but only a few will be accepted by highly-selective schools. In the latter case, nearly every graduating student is headed for the Ivy League or some other "elite" institution.