If you have savings then they will assume that you can spend them on education. If you don't they will pay (the end cost for me was roughly the same as private school fees in the UK). Until recently I didn't think the lesser Unis did this but it transpires they do (I thought they only offered tiny scholarships)
Let me see if I can clarify this a bit.
There are two categories of financial aid, need based aid and "merit" aid. Need based aid is pretty straight forward. They look at student assets, parental assets and income and calculate an Expected Family Contribution (EFC). If your EFC exceeds the cost of attending you get no need based aid. If your EFC is less than the cost of attending, you are expected to take out the federal government loans on offer, which don't add up to all that much. Then the university will fund the rest with a grant and in some cases, a part time job on campus. University money is always "the last dollar in" as they say. I don't know what they do with foreign students who obviously don't qualify for US federal loans.
The EFC can be really big. We're sort of upper middle class and our EFC was over $70k, so no need based aid for DS. He still had the option of taking out an unsubsidized federal loan, which we managed to avoid. When DD starts in 2020 and we have two attending at the same time the EFC for each will be $35k, but then the kids would have to take out loans for the next layer of the cost, and we'd like to avoid that if possible.
Merit aid is much simpler. Uni's are ranked by all sorts of services and publications and a big part of the ranking is the average Scholastic Aptitude Test (SAT) scores of the students who attend. The SAT is given in secondary school and has two components, English and maths. Each section is scored from 400 to 800 and the results added together for a total score. A total score of over 1200 puts you in the upper half of the results.
Universities seek students whose SAT scores will raise their average and thereby improve their ranking. They effectively "buy" high scoring students by offering them merit aid packages that make attending more affordable. So less selective uni's will offer a high scoring student a lot of money. There are other factors as well, like athletics or musical ability, etc. that can increase your merit aid offer. The merit aid will be taken away if your grades fall below a certain level, then restored when your grades come back up.
I read an interview with a university admissions officer who likened the process to casting a movie. You eliminate those who are clearly unqualified then have to bring in a certain number of football players, German students, and the odd French Horn player.
We were fairly fortunate in that DS had very solid SAT scores and is an accomplished crew rower. His uni is giving him $12k per year in merit aid, which brings his total cost per year (tuition, room and board, fees, etc.) to about $40k per year. Other, less selective schools offer about double that, but didn't offer as good a program or facilities as the one he now attends.
Part of what is driving the competition is a demographic drop in the number of university aged kids in the US. There are about 300k fewer of them then there were 5-10 years ago. The really top schools aren't affected but the mid range ones find themselves really competing more sharply for the students they want. Quite a few lower tier uni's have either closed or are downsizing and laying off faculty, closing dormitories, etc.
If you want to lower the cost of attending a university in the US it's pretty easy. Have your child apply to schools where they will be in the upper third of the applicant pool. You will very likely get substantial merit aid if the uni offers it, and most do.
Sorry this is so long. I'm happy to answer any questions. 