I just want to point out that poor student are absolutely not at the mercy of rich students. Poor students are needed at these schools to maintain the prestige

I just want to point out that poor student are absolutely not at the mercy of rich students. Poor students are needed at these schools to maintain the prestige and reputation of these schools. If they only allowed those who could afford to pay, a lot of great talent would end up elsewhere and then the reputation of the school would suffer. In terms of real scarcity, it is actually the rich students who are the most easily replaceable. If those poor students went somewhere else, they would still do well and within a decade the new schools would start challenging the reputation of these elite school.