The causal chain seems to roughly be: Top jobs tend to belong to those who attended top universities. => Top universities are attended by those who went to private schools. => Those who went to private school are either from wealthy families or attending on scholarship due largely to the fortune of genetics and/or an educated family. I certainly benefited from that (school scholarship => Oxbridge => good tech job), but I don't know that I could have if fortune hadn't smiled on me.
I agree that even with all those benefits, you can still fail if you make poor choices. However, I don't think the vast majority of people with worse circumstances can just choose their way out of it. It's not to say that it's impossible - my grandma has a rags to riches story that's only slightly marred by her similarly ethically dubious sales techniques...