You're just as likely to have bureaucracy screw up things in a "private" (which normally means "corporate") system as in a public one. It's quite possible to organise things in a system of common ownership without falling into that trap, and it's how communities all over the world operated for thousands of years.

As for health, it's always seemed to me fundamental that whatever incentives there are to encourage the health workers to do a better job should be based round keeping people healthy rather than treating them when they are sick. That is always being claimed as the underlying principle in traditional Chinese medicine - I don't know if that's true or not, but it makes sense.