I suppose it depends on area in which you live, but my local private schools in the Bay Area cost $45K, $41K, $44K, $40K, and $44K. I realize the Bay Area is expensive.
Top private universities are biased against rich people, simply because they already have a lot of them. Obviously, if you attend the university, you will meet lots of rich people who got in. You don't get the chance to meet the ones who didn't. Rich people are expected to give money, but they don't necessarily tell people, or their children, that they do.
2350, which is what I guess only 3000 people manage on the SATs, is high, but Stanford, for example, has 54%, 42% and 54% of people scoring below 700 on Writing, Math, and Reading. High SAT scores are not enough to be admitted, if you are from a privileged background. The schools make allowances for your background and require higher standards. Realistically, this is fair, as richer people have more access to tutors etc., but as you bump up against the limits of grades it makes it hard to distinguish really great from great, so rich kids are required to have other characteristics in order to be admitted.
Colleges might not have the choice of an actually poor (lowest 20%) child with similar grades, but they do have the choice of a middle class child with grades that are within the error bar, and for reasons which can be justified, they choose the middle class child.
I know quite a few children of the 0.1% in Ivy League colleges, and, almost without exception, they are there on sports scholarships or they made significant donations. I don't know how much each person donated, except in certain cases, like the $1M for 5 years I mentioned. That person is quite rich, so that number might be high.