Earlier quoted context omitted.
Have you been on a university hiring committee? My anecdotal evidence is that yes prestige very strongly matters and coming from a very prestigious university often outweights other factors such as publication record. I haven't read the original article yet but one can easily test if e.g. the publication record of a graduate student from a non-and prestigious uni differ to the extend that is indicated by the hiring.
All else being equal eg publication record, a candidate A from a less prestigious place is ranked higher than candidate B from a more prestigious place. This is because the candidate A did the same stuff with much less support than B. The issue is that prestige is highly correlated to research outputs, so it is difficult to disentangle.
In my experience if you have done a postdoc/PhD at e.g. Stanford people will look much less thoroughly at your publication record. I've actually argued in committees that a candidate from a less prestigious school should be reranked higher than a candidate from a prestigious school, because they had a much better publication record. The interesting thing was once I brought this up in the discussion, most people agreed. However in their own ranking they had put the person from the prestigious school higher, which indicates to me that this is an unconscious bias.