I find this interesting. My team at a large public university in the United States just failed two rounds of interviews for a senior full stack software engineer position. Salary is around 130k full compensation. This number includes fantastic benefits and 12% match for 401k . We failed the interview because the candidates that we get are very inexperienced and they interviewed terribly not knowing even basic things. However we can't seem to get anyone more experience to apply because their salary demands are too high, so we end up with entry level engineers attempting to apply to senior roles. The university can't afford to pay anymore so we're kind of stuck.
My wife works for a private university (you've heard of it), in a non-technical role, and it's the same there according to her and her friends at work. The pay is mediocre at best, but the benefits kick ass. Those include excellent health insurance, which as most reading this probably know is a very big consideration in the USA. So maybe that's just how universities do things.
Depending on where said fancy private uni that doesn't pay well is located might be an issue. Yes perks are good. But mediocre pay only goes so far if you're living in a high tax, high rent state.