With one thing to think about. Stanford offers Schollies but last I knew most Ivy league schools do not.
Eh, yes and no. The Ivies don't officially offer athletic scholarships, but they have millions of dollars in endowments that are available for various "leadership grants," which just happen to be distributed to those who go out for sports. It's a facade, and it's hypocritical, but they can pat themselves on the back for NOT buying into the high dollar grubby business of D1 college football while basically doing the same thing with a different label. Ben Sasse received grants from Harvard that related to him going out for wrestling, and he got some sort of slight bump for trying out for football. He wasn't special in that regard. It's the norm as long as you're not from a wealthy family.
They offer neither academic or athletic scholarships
That might be true for some schools--I'm not sure--but I'm not aware of any major U.S. university that doesn't offer academic scholarships. Some don't offer full scholarships, but these things are often a shell game where what looks like a scholarship in practical terms is called a "leadership grant" or something similar. A lot of these universities--pretty much all Ivies, plus Stanford, Duke, University of Chicago, Johns Hopkins, et al.--don't need to charge tuition at all, but they typically charge tuition from those who can afford it. They often charge incredibly high rates of tuition because they can, and because people are willing to pay for it. That revenue gets shuffled around to cover those who don't pay, but it's almost all surplus, regardless.
Most of them basically charge on a curve. If you are anywhere near lower income, you can go there with a big chunk of the cost paid by the University. The full ticket price is set to maximize the cash they take from high income parents and from student loans, etc.
This is spot on. Even local private schools do this. The colloquial term used for those students who are wealthy enough that they don't qualify for any grants, etc., is that they're "paying full freight." Foreign students and marginal students from wealthy families pay full freight. Because Pell grants and Stafford loans are based on a combination of the price of tuition and the ability to pay of the student (and student's family), there is a perverse incentive to jack up the rates as high as possible so as to rake in as much money as possible. Whenever the Pell or Stafford numbers go up (meaning what can be paid per year per student), coincidentally the universities and colleges jack up their rates by almost the exact same amount.
Getting in is the hard part. Once you're in, then they can be generous in making sure you can figure out how to pay for it.
This is also true, but this varies by background. When I taught in Kabul, my Afghan students were put to the front of the line for grants, etc., that were simply not available to a middle-class American student. Vice versa, Harvard and other Ivies have a disproportionate number of Korean, Taiwanese, Singaporean, and Chinese students because a) they got the necessary scores on their SATs to qualify, and B) their parents are able to pay full freight. Wealthy international students are the cash cow that keeps on paying private universities, and they're a major reason why the prices keep going up ... for them at least.
Back to my Afghan students, the first graduating class from the American high school in Kabul had an Afghan female valedictorian who was incredibly smart and hard-working, and her test scores were home runs. Because of the combination of her being female and from such a difficult culture/society/country, she had her choice of universities to attend in the U.S. for free. Most public universities had limits as to what they could offer, but the private schools didn't. Due to a combination of proximity to people she knew (local Afghans living in the school's area) and the full scholarship+ that she received, she ended up going to the University of Richmond for free (and even earned some stipends). The University of Richmond officially has not ever given a full-ride academic scholarship before, but they wanted her that bad. She graduated magna cum laude with majors in economics, German, and something else, which was what everybody wanted.
Ben Sasse repeatedly referred to his admission to Harvard as being part of their "hayseed scholarships" for people from what they now call "flyover country." His grades, test scores, etc., were off the charts, but he probably wouldn't have been admitted if he had identical scores, etc., but grew up in Connecticut in a family that made $250,000 per year. I pick that income on purpose because it's too high to receive aid, and too low to likely be a potential big donor.
I was working at a private college in the region when they went through the process of dropping down from D2 to NAIA so that they could increase revenues through athletic scholarships. If that seems counter-intuitive, it's because you assume that everybody else is logical and makes wise decisions regarding money; you're adorable. Schools jack up their rates so that they can then offer meaningless small scholarships to student-athletes who want to be able to brag to their friends that they're "on scholarship" at such-and-such university. It's kind of like how it seems like everybody comes back from a trip to Vegas talking about how they didn't lose any money, yet those casinos keep getting bigger. The school where I was working raised their tuition by $8,000 but then began offering a mind-numbing number of $5,000 partial scholarships for every sport imaginable (including their newly minted competitive dance team), and new students lined up like sheep to be fleeced. Think of it as a vanity scholarship: Mom & Dad get to brag that little Jimmy is on scholarship at Cooler Than U, and they pay a few thousand dollars extra for the privilege of saying so. Welcome to America.
That's how the sausage is made.