But the private schools are not the ones to blame

I'm not sure if i remember correctly where you are from, so correct and forgive me if my guess is wrong, but i assume you mean the US.
The reason american leaders come from private schools is because in the US, only a particularily wealthy man has a chance at going far in politics. Some of these gained theyr wealth, some inherited it. In the former case, the private school possibly plaied a positive role, in that it gave the kid what it takes to succeed in this world {i'm not saying it means it upheld the child's nature, only that it did atleast something good}, in the latter, it had no relevant mening at all, because wealthy parrents generally send theyr heirs to private schools, which are commmonly percieved as better, regardless of whether they really are.