One could look if it made any difference, but the reason for what they see might not be obvious to understand. The obvious expected change would be an increase of the quality metrics and a decrease of the (previous) prestige metrics in correlation of which service customers buy. (In principle, this is calculable.)
In practice, I'd expect the effects would be lower than “rational”, at least at first. Such a precise quality metric is likely to be complicated to obtain (since we don't have one yet). People would naturally be mistrustful of it; in effect, unless one can do the entire analysis again and re-derive the new metric themselves, one must balance the previous prestige measure with the prestige of the researcher(s) that introduced the new metric. (In fact, even if I could re-derive the metric myself, I wouldn't apply 100% confidence to my calculations. In all psychological studies that I read researchers seemed to assume that the rational action would be to apply 100% confidence to new information, with the exception of studies about confidence...)
The effect might be even reversed, in the case of more-or-less paranoid customers suspecting a conspiracy.
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.