If we talked of the poor the way we talk about the 'rich' we would find ourselves hounded to the ends of the earth. Anyway, what is a definition of rich - twice the average wage, 4 times, 6 times, what?
So 7% of the population went to private schools get 39% of the jobs, but but that is a lot less than half and the other 61% of people in power presumably come from homes that are not 'rich and privileged.
It is the same with Oxford and Cambridge well over 50% of students at those two universities are from state schools. If 24 percent of their graduates get top jobs, it could be because they are actually a damn sight brighter than the rest of the population. I mean they are elite universities and you do have to be both very bright and very rounded to get in there. Presumably most of that 24% would be state educated.
I have very little time for those who winge on constantly about privilege. My grandparents, one a docker married to the daughter of Irish immigrants living in a slum in London, the other an Irish factory worker, who enlisted in the army and married a shop assistant, all started life in poverty and and near the bottom socially. All died as members of the comfortable 'middle class'. One grandfather became an army officer decorated by the queen and the pope for his charitable work.
They got there by being clever and socially ambitious for themselves and their children. No privilege, no strings pulled. That way has always been open for the determined and ambitious and it still is. My FiL worked on the assembly line at Vauxhall. His son, my DH, got to grammar school, university and proved so good at what he does, that he is still being called back to advise on major engineering projects in his mid 70s.