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. Obviously it's easy for a bunch of rich kids in small classes in communities where they are all pushed for academic excellence to excel.Where I live in the VA and MD Washington Suburbs within most of the school districts many of the public schools are actually better than anything the private one's could offer.
For example, most of the high schools within Arlington, Fairfax, and Loudoun County VA or Montgomery/Howard Counties MD, are ranked higher or on par then most private schools in the area.
Also, the high school I actually attended was ranked in like the like the top 50 schools overall in the nation.