I would argue that schools don't do this because it's in their best interest to admit a certain percentage of kids who can't afford it, and bend over backwards

I would argue that schools don't do this because it's in their best interest to admit a certain percentage of kids who can't afford it, and bend over backwards to make sure the kids attend. If the schools fill up with ONLY rich kids with excellent academics, they're going to miss out on the poor (or middle class) kids who will clearly go on to do great things. The output will get bland, and they're leaving a huge market gap open for another educational institution to come in and sweep up these poor geniuses and build a reputation on that. You need to strike a delicate balance between the Malia Obamas, the Rothschilds, and the Ramanujans out there in order to build and maintain the reputation of the institution. Even if they're losing money, and dealing with a paperwork nightmare in the short term, it's a long term gain for them. So the schools don't want to eliminate these loans and subsidy programs, the consumers certainly don't want to eliminate them, and because of this, the government doesn't want to eliminate them -- adding more subsidies, scholarships, and increasing access to loans is an extremely popular political move. I would also argue that these subsidies provide a net economic benefit to society, by allocating educational resources to those who will get the most out of it, but that's a little more controversial.