Nope, I don't think that has anything to do with it. Harvard's policy is pretty similar to Princeton's described by hkmurakami. You can check out the calculations at [1], but the basic idea is that it's free up to 60k family income for Americans regardless of race then about 10% up to 180k income. This means that you can be making the median family income and still have an expected family contribution of $0 at Harvard. Obviously these sorts of levels of financial aid are only at the top few schools, but at places like Harvard and Princeton it really is independent of race.
One of the issues plaguing the UK is that the wealthy students go to private schools, which have the best facilities/teachers etc and are able to produce far far higher results. The argument becomes "when n% of the people with maximum A level grades come from private schools, that's the % you should expect at the top universities which gate by grades, regardless of their history".
So I guess I have 2 questions:
Do you have a similar issue where the wealthy buy better tuition to ensure higher results and get a better shot at the best colleges?
Is there a scaling factor applied to those of different backgrounds to mitigate this?
I think the answers to those two questions are roughly yes and yes. I get the impression a lower percentage of people here go to private schools in general (and could be totally wrong), but many do. There are also plenty of other opportunities that are easier to access with wealth. Even with public schools, wealthier people tend to live in school districts with better schools.
There's no simple scaling factor because there's no numerical value to scale (admissions at places like Harvard are not just an objective formula of grades and test scores) but background is certainly taken into account. A kid going to a school without lots of AP classes isn't expected to take lots of AP classes, for example. A kid from a rough economic background who has to work a minimum wage summer job for financial reasons will get more understanding than the rich kid who chooses a summer job for spending money instead of volunteering in a research lab, and so on.
As I mentioned in another comment, I think one challenge is that it is harder to differentiate kids without these opportunities. To use the AP class example, you can help separate kids from top schools by looking at how many/which AP classes they took and what scores they got on the tests. If you take a similarly sized cohort of kids at a school without AP classes, how do you differentiate them?
[1] https://college.harvard.edu/financial-aid/net-price-calculat...