Also British living in the USA. For us instate would have been cheapest but this depends on which state you live in. Overseas fees in the UK were not doable for eldest (especially as they were competitive for highly ranked universities and Oxbridge and London universities have ridiculous overseas fees) and they are at a liberal arts college that offered them a generous package.
The biggest source of funding for private universities is need based aid. If you are well off, then you are unlikely to benefit. There is a second problem, which is whether need based aid is distributed to international students, and if admissions are need blind for them.
state universities typically don't meet full need for anyone and consequently don't look at your finances when they offer admissions. it's all on you.
Sports can mean one of two things depending. It can mean a specific scholarship, or it can be an admissions boost but with no scholarship attached. If a college is Division III they are not allowed to offer scholarships, but you will find that students are still recruited and may get in with not quite as strong an academic record. I believe. The ivies do not have designated sports scholarships, but they do recruit. The money part at the ivies is all need based. If your family is wealthy you're going to pay, it's quite simple for them.
(don't forget that the Ivy League is a historical association so there absolutely are other universities just as good!)
not applicable for OP but I wouldn't take a scholarship to a lesser ranked university if my intention were to return to the UK, unless it were a really fabulous deal or the child were mad about sport. It cuts both ways. There are well ranked British universities that a lot of Americans don't know at all. Someone who is well clued in to economics would recognise Warwick for example, but I couldn't say that an ordinary HR manager would.