Do private schools have as much of an influence over american businesses/government?

in the uk we a huge proportion of privately educated people hold the highest roles. unis that have lots of private school students, compared to only 7% of kids being in one. My brother and I were both educated at one of the "famous" private schools. But our dad was working class (in the british sense), so always made us aware of our privilege. Whereas our mum is very wealthy. I feel like we grew up with both sides. A big discussion in england is that most of our corporations and government are ran by privately educated people. we also have a culture of these "famous private schools". I have family in america but they all went to state schools instead of private. I was wondering if this same ideology or culture exists in the US?