The problem is the rules can be broken with minimal consequence. Swap out PE for another profit-seeking structure and you’ll tend towards the same outcome, as the bad outcompetes the good.
The optimum might be an employee-owned facility, but even there you'd have incentives to increase profit - everyone would like to get paid more for whatever it is they do. PE has the strongest conflict of interest though, as they are simply investors seeking profit and have nothing else in the game.
This is the red herring. PE is one of many categories of financial investors. (Worse, it has become incredibly nebulously defined.)
I'd also argue that employee-owned coöperatives might be the single structure worse for healthcare than a rapacious, distant capitalist. The latter can fuck around with the books. The former can fuck with the charts.
In a similar vein, even as a libertarian minded person, I don't like the idea of private prisons at all, I don't feel a company should be incentivized towards keeping people locked up and suspended of their rights. Not that I don't believe in prisons, only that they shouldn't be businesses.
I think with medical facilities, that there should be far less protections from liability, and more severe repercussions from any coverup, non-reporting or mis-reporting of harm/accident/injury. Not to mention more, regular inspections and audits, along with camera/recording requirements.
Seems like a big part of their bread and butter is anti-competitive or otherwise legally questionable self dealing between their holdings. Things that would quickly face public scrutiny if those companies were publicly held and had the associated reporting requirements.
This is all private ownership, including non-profit.