> The research question here is whether faculty are hired from those prestigious schools above and beyond the rate at which they would have been hired based on other signals of their potential as researchers

It's interesting that you use the word "potential". Tenure-track positions are very hard to get, and most newly minded PhDs have very little track record. If they're lucky, they have a publication or two in a journal, but many will have only their dissertation. So I think the question is whether PhDs coming from prestigious schools are judged to have more "research potential" based on where they come from rather than their limited record.

The "potential" problem is even worse when it comes to admitting undergraduates into graduate programs.

If you don't get a tenure-track position right out of grad school, it can be difficult to ever work your way up to one, because you'll probably have to take a job at a school with a greater teaching load than a typical research university, which leaves little time for you to do your research and prove yourself. In a sense, the "potential" becomes a self-fulfilling prophesy.