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.