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.
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.