But... they already do.
I can't find links right now, but deans of admissions from Harvard, Yale, Princeton, etc. all talk about how they could admit 10x the students and they'd all be equally good.
And that their selections aren't exactly "random" but are focused on diversity: that there are enough players for each instrument in the orchestra, athletes for each sport, varied extracurriculars in general, students for each major, with a diversity of race, ethnicity, nationality, geography, and so on. That early-decision candidates obviously have a bigger chance, etc.
But for a student who gets into Harvard and Princeton but not Yale or Columbia, it is essentially random -- e.g. Harvard and Princeton hadn't allotted all their trumpet players already but Yale and Columbia did.