There's a contrarian theory presented by Robin that people go to highly reputable schools, visit highly reputable hospitals, buy highly reputable brands etc. to affiliate with high status individuals and institutions.
But what would a person who completely didn't care about such affiliations do? Pretty much the same thing. Unless you know a lot about schools, hospitals, and everything else, you're better off simply following prestige as proxy for quality (in addition to price and all the other usual criteria). There's no denying that prestige is better indicator of quality than random chance - the question is - is it the best we can do?
It's possible to come up with alternative measures, which might correlate with quality too, like operation success rates for hospitals, graduation rates for schools etc. But if they really indicated quality that well, wouldn't they be simply included in institution's prestige, and lose their predictive status? The argument is highly analogous to one for efficient market hypothesis (or to some extent with Bayesian beauty contest with schools, as prestige might indicate quality of other students). Very often there are severe faults with alternative measures, like with operation success rates without correcting for patient demographics.
If you postulate that you have better indicator of quality than prestige, you need to do some explaining. Why is it not included in prestige already? I don't propose any magical thinking about prestige, but we shouldn't be as eager to throw it away completely as some seem to be.
I hate to be "that guy" but it depends on what you mean by quality! I think what you have in mind is something like, does the school educate you well? But the only way you'd know if it educated you well is if you succeed in life, i.e. achieved status. ("No, standardized tests!" -- but how the questions on the test are deemed important is irrevocably intertwined with status to begin with!) So we see that education is a tool for achieving status. It's much more direct if the school just conferred status -- which is what we find.
So yes! Prestige is a great indicator of quality - as long as you define "quality" as "prestigious"! (Which we mostly do anyway.)
If that's not clear (it's hard for me to get ideas out of my head sometimes), I guess another way to say it is: Which causes the other? Because I think "prestige" is an input to quality, whereas you're saying "quality" is an input to "prestige."