You'd think the medical schools might remain independent under their universities, but that often turns out to not be the case as well. Turns out asking for handouts from the endowment looks a lot like bankruptcy, and the rest of the university takes a dim view of it, especially considering how home addresses of their physician colleagues (hint: very large houses in the best neighborhoods). See Tulane, Georgetown, etc. The state school systems, like the UCs, tend to do better, or are at least shielded by the state budget.
If you point to Mayo, Stanford, and Partners as exceptions, I'd say they're the exceptions that prove the rule. Mayo has always been a self-sustaining business. Stanford and Partners have ready consultation with the strongest business schools in the US, if not the world, and the endowments to match.
On the normative question of blaming groups, it’s like that imposter Spider-Man meme of each person pointing at the other. Hospitals charge 10x the actual market price and health insurance boasts on your explanation of benefits that they got you a 90% discount. Americans demand cutting edge maximum intervention perfectly safe treatment with no formal government rationing, or HMO rationing.
Health care is a for profit sector. Fees are paid by third parties (so patients are not directly exposed to routine costs). There is heavy quality regulation. Providers are incentivized to treat more through fee for service so the sicker you get and the more procedures they perform the more revenue they earn (perverse incentive). There is no counterweight to negotiate prices down. The end result is 7% medical inflation per year.