There really isn't such a mismatch between aspirations and money available when it comes to universities in the US.
The less well off do pretty well in the American university picture. You're not going to get into a good university if your grades are low, which is understandable and not controversial. But if you are able to do well in high school, you have a good chance of going to a well-regarded university and paying very little for your degree.
And there are routes to university through community colleges, and ROTC, where the armed forces foot your bill at a state university, under the terms of the GI Bill, and you are commissioned as an officer upon graduation - you then serve an appointed number of years and can retire and move into the private sector or pu lic service (your armed forces service is very attractive in the public sector) with your degree and your active duty officer experience under your belt. You join the reserves and do periodic training, and you are entitled to VA benefits and an officer's pension on top of whatever other pension or retirement provision you get from your career outside the forces. DS' roommate in his final year was doing a degree in French, paid for by the US government. He was a second generation American of Central American heritage.
The endowment problem that British universities suffer (honorable exceptions noted) is going to be a huge issue now that EU research money isn't available anymore. Research opportunities attract top academic talent. If the money for research isn't available from outside sources (government, EU, industry) and there isn't enough funding available from an institution itself to support research (endowment problem) there will be an increasing stratification of British universities, with a select few at the top and the rest a lot further down, caught in a vicious cycle caused by funding issues.
There's a reason why all those British kids who pull themselves up by their bootstraps end up doing PhDs in the US. It's money.