And hiring committees also. It is this: the fact that x has received better training than y - gone to a better school, worked with more accomplished teachers, etc. - is a reason, cp, to prefer y to x.
All this is, in my view, completely irrational. I will grant that treating quality of training as a positive feature of an application has some limited rational utility, if treated as a highly defeasible principle, at very early stages. If we see that x has strong letters from a school that trains people well, that is perhaps a stronger reason to send them to the second round than is the fact that y has strong letters from less impressive folks. But once we are at the stage of evaluating actual first-order evidence - that is, reading papers - things switch. If x has produced a paper of quality q without great training, and with little help from world-class teachers, while y has done the same with these benefits, this obviously speaks in favor of x. y is already, we might reasonably think, performing at her best. After all, she is getting the best help there is. But x, well, x could really take off when she gets into a better context. So place your bets on x if you have to use this factor at all. (Of course one might be thinking: "well, x may be better than y, but less well trained, so we'd have to do more work to get them up to speed." That's an understandable, but I hope obviously not rationally endorseable, consideration. Lots of other disreputable, but understandable, motives can be imagined such as wanting to please famous people in the field. I ignore all this and assume virtue on the part of committees.)
I think this is just clear. And not just as some tiny tie-breaker that should matter in virtually no cases. I think it is a significant, substantive reason to prefer someone. If we are talking job candidates, the search process will move to a stage of intensive scrutiny such that this consideration should probably drop out. If a search committee is doing their job, they will have read multiple papers, talked with the candidate at length, seen them present, etc. Probably at that point, it is reasonable to make all decisions based on first-hand considerations. So for job searches I'm inclined to think that quality of grad school is most important at the middle stage, the one where we narrow from plausible candidates to interviews. But for admissions, there is no final stage. Decisions are made on the basis of reading one, or at most two papers. And at this stage, quality of training is an important factor, just one that cuts in the opposite direction to how it is usually taken.