Bottom line there are a lot of very incompetently run facilities out there, of both private, "private equity" and various form of public ownership/management.
It makes a great headline to say that a hospital can be "bought" but ownership and management of hospital systems takes a book to explain. Outside entirely vertical systems like Kaiser and even that has some complexity, hospitals are much more like medical malls than anything else. An ownership and or management company does not dictate what happens under their roofs in anything like the manner a private corporation like Coke, GE or Google would. 29 states make it outright illegal for non-licensed persons to own or manage an entity that engages in the practice or medicine. Most hospitals of any size over 100 beds involve at least dozens of seperately controled and managed entities. See "corporate practice of medicine (CPOM)" laws.
Finally I would add that measuring outcomes on an apples to apples basis is more or less an impossible thing to do. Lots of people don't like to hear that. Again something like HAI I think can be a useful general indicator as well as other measurements of preventable complications, but comparing outcomes at different facilities especially across size and location is more or less useless.