Not to show teeth, but I can
assure you no Stanford alums are "jealous" of USC, and
we often refer to it as U$C or University of Spoiled Children. And while many great people go to USC, and many of them are not spoiled children, the ethos of the institution definitely has a "money talks" vibe. Indeed, your own description of the institution exhibits a money first attitude. And isn't the fact that a bunch of spoiled kids want to buy their way into USC,
and often do, suggestive of the fact that the moniker is rather apt?
One recent ignoble instance was the Dean of the Medical School, who was a known substance abuser caught in a hotel with a prostitute who had OD'd. For over a year, the incident was hushed, because he was a prolific fundraiser.
I could share other stories, but LAist has compiled a nice list of them here...
ETA: And I should say I have several relatives who are U$C alums, so this definitely comes with the (quasi)friendly intra-family rivalry territory.