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.
There seem to exist certain measures of quality that are second level, in the sense that they measure quality in a kind of indirect way, mostly because the indirect way seems to be easier. One example is sex appeal. The "quality" of a potential mate should be measured just by the number of healthy offsprings it can give birth to. However, that's difficult to find out and hence evolution has programmed our genes to refer to the sex appeal instead, that is, the number of people who will find the person in question attractive. However, the only problem with such second level measures is that they can be feigned. An impotent person can still look attractive.
Similarly, imagine a car manufacturing company that produces two different models. However, the only differences between them are in the looks and the cost. Assuming that the looks are not clearly better than the looks of the other, people who use prestige as the measure of quality will always buy the more expensive car.
Modified, naturally, by how sexy the healthy children are themselves.