I think the right solution is strict test-based admissions, like in Sweden, and forbidding schools from admitting other than on test results. In addition to that, making things like getting top government jobs also test based. In Sweden we do this for diplomats. You have to get a university degree, but whether you get in doesn't depend primarily on prestige or interviews, it depends on whether you pass a somewhat difficult test.
In this way you ensure that what university you went to doesn't really matter, and over time what I imagine is a more distributed university system where co-operation with institutions and someone's particular ideas matters more than what institutions he's associated with, and where the extreme prestige of particular schools disappears.
I think it's especially important to eliminate the hoop jumping, so that people can know that, as long as the extreme prestige universities exist, they can get into them if they perform well enough on exams, with nothing being able to interfere with that-- no individual judgement, no subjectivity, no hoop you haven't jumped. Just pure merit, like the Swedish or French system.
> her fellow students were claiming they were disabled to receive accommodations like extra time on tests, excused absences and the best housing on campus
As far as the problem you're talking about, I get what you're describing and agree it sounds appealing. The problem in the US is the university hierarchy is one of the things used to perpetuate class, so the people who are able to change it generally benefit from how it currently is.