To respond to the first part of your comment, obviously "what a person can afford to pay" is subjective, and lots of families still don't find their education "cheap" (I have a friend who got roughly $51k of financial aid on a $52k bill, so his family's contribution was only $1k/yr - they still struggled to pay that). But most top schools don't expect students to take out student loans of any form (see this list on wikipedia), and I think most poor / middle-class families will find that their tuition bill at an elite school is smaller than it would be at a good state school; for example Berkeley tuition is $13k/yr for in-state students (not including room and board) and more than double that for out-of-state/international students, while Stanford just down the street has free tuition for any family earning less than $100k/yr. So again, you can argue that elite schools aren't the most needy targets for charitable donations (do we really need to help Stanford extend their free tuition offer up to families earning $150k?), but I don't think it's fair to say that they're actively trying to bleed their students.

I can't speak for Princeton directly, but I think top-tier university finances and top-tier LAC finances are similar enough that my experience at Williams might be relevant. Williams spends about $80k per student per year (roughly a $160M budget divided by 2k students). Nominal tuition is about $50k including room and board, and financial aid discounting means that the average student pays about $30k (some students pay the full $50k, but others pay nothing, and most pay somewhere in between). So the average student is getting about $50k spent on them that they're not paying for; that difference is funded primarily from endowment returns. I think this situation is similar at Harvard/Princeton/MIT - even full-fare students are paying less than the total amount that's being spent on them.

Where does that money go? At Williams it mostly goes towards people: a 7:1 student-faculty ratio and something like a 2.5:1 staff-student ratio. Williams employs 800 staff for 2000 students, which sounds ridiculous and probably is pretty ridiculous, but that includes everything from chefs and dining employees, to secretaries and departmental administrative assistants, to IT support, to mechanics and janitors and groundskeepers, to doctors and nurses and psych counselors and chaplains, to athletic coaches, to campus life coordinators and deans. I'm now in grad school at a much larger and less-well-funded state school (UC Berkeley), and it's amazing to see the contrast between the two institutions. A lot of kids get decent educations at Berkeley, and some do really well. But where at Williams I was taking classes with 20, 10, or even 2 students in them, Berkeley undergrad class sizes are in the hundreds. At Williams I was working on a research project with a prof by the end of my freshman year; at Berkeley most undergrads don't ever get a chance to get involved with research. And so on. You just don't get the same level of personal attention at a large state school (and Berkeley is one of the best large state schools!) as you do at a small private school. That level of attention is expensive to provide.

I've seen the studies about educational outcomes at elite schools before, and I believe them to a certain extent, but you have to be at least a bit careful interpreting their conclusions. I'm pretty sure I would have ended up "successful" by most metrics if I'd gone to a large state school like Berkeley, in the sense that I would have gotten decent grades and graduated and found a decent-paying job somewhere that I would have been happy with. But I probably wouldn't have gotten involved in research and ended up at a top grad school. I probably wouldn't have met nearly as diverse a group of friends from all around the country and around the world. I probably wouldn't have gotten to meet and have lengthy conversations with people like Dan Dennett and Steve Strogatz, and I probably wouldn't have had a whole bunch of other experiences that have made my life richer in ways that are not going to be measured by a few survey questions as to my "success".

You can certainly get a great education at a lot of schools (heck, if you're the right sort of student you can get a great education sitting at home with a textbook), though I dispute that your average "Springfield State School" has anywhere near the same level of amazing people and opportunities as a top-tier university (case in point: I was going to check out the course offerings of the SUNY Cortland computer science department to see how they compared to CS programs I'm familiar with, but it looks like they don't actually have a computer science department, just a minor in "computer applications"). But if you're lucky enough to be able to get into one of the elite schools, there are experiences you'll be able to get there that you just won't have at your average state school. I don't know if supporting elite schools is the best use of my money as a potential alumni donor, because obviously there are a lot of other worthy causes in the world. But I do think that the world is a better place for having institutions whose mission is to bring together some of the best students from all around the world and give them the best educations money can buy, so I certainly respect people who choose to support that mission financially.

Some schools might. But as I said, the top tier of American universities are need-blind and guarantee to meet full financial need for all students, domestic and international. One of my best friends from college is from Bangalore, not from the slums but certainly not a rich family, and he paid $0 in tuition for four years at the number-one-ranked undergrad school in the US (Williams College, which has nominal tuition of $50k/yr, but which guarantees to meet full need for all students and is just a couple of technicalities away from being the seventh entry on my list above), so it does happen. And it's not even a rare thing; the majority of my international student friends paid little or no tuition (and of course, neither do many domestic students).

People don't understand that in the top tier of American education, it's not about money. Harvard is swimming in so much money they've actually seriously debated whether they should eliminate tuition entirely and just become free for all applicants; the only reason they haven't done this is that they're already free or very cheap for basically any middle-class family, so the only people who would benefit from such a change are the people who are rich enough to not need it. MIT is not quite that rich, but they're rich enough to offer very generous aid to any international student they admit. And that's exactly what they do, because their reputation as one of the best schools in the world is very much dependent on their having the best students in the world, and that means making sure people can come regardless of their ability to pay.

Just curious, what school did you go to? My experience from friends at wide range of schools is that there are a lot of mid-tier American schools which do make an effort to squeeze their students dry, but the schools at the very top tend to be wealthy enough (thanks to alumni donations) that they can afford to have very generous financial aid policies. MIT in particular is one of only six American schools that do need-blind admissions for all students and which guarantee to meet full financial need (the others are Dartmouth, Harvard, Princeton, Yale and Amherst). That means that no one at MIT looks at how much money you have when they decide whether to admit you, and once you're admitted you're guaranteed whatever financial aid is necessary to allow you to attend based on your family's financial documents. I think MIT's financial aid policies are slightly less generous than, say, Harvard, which is actually free if your family earns under $60k/yr, but that's only because Harvard's endowment is more than three times as large.

Basically, the argument that schools like MIT make to their alumni is that when you donate to MIT, you're not giving your money to a large faceless entity with a $8 billion endowment staffed with administrators who light their cigars with $100 bills. You're giving it to a poor kid from the slums of Bangalore who is able to come to MIT and fulfill his/her potential because of generous alumni like you, who have allowed MIT to provide a $300k education for free to anyone who can qualify. Obviously you can believe this sort of thing to varying degrees, but apparently Bose's experiences working at MIT for several decades came to convince him that it is, as an institution, overall a force for good in the world.