Yeah that sounds fine too if you can do it. Anything that prevents hospitals from being operated by large profit-seeking entities with no other stake in them would probably be a strict improvement over the current system.
I think the practical issue is those fields that have similar restrictions basically predate a major societal shift. We now consider the only valid limits on profit and ambition to be market forces. I'm not sure restricting hospitals in this way is less radical than just nationalizing them, in terms of practical politics.
Anyway, again, sure. I'm not informed enough on this subject to know what model would actually work best. I think the problem is the raw exclusive profit motive rather than who specifically is running them, but there are a lot of ways to eliminate that.