The mixed economy works wonders here in Denmark - Government cutting costs isn't an issue for long since the media and people watch over it. There are blunders and problems (Moreso with liberalconservative governments but no surprise there) here and there but for the most part the system works.
A note on motivation:
Governments supply these things because the people want them to
Private companies supply them because they can get money for it.
Logic dictates that private companies have more interest in cutting corners or undertreating/rejecting patients since they are the only ones who get something out of doing so.
Also this should probably go in the politics forum so to avoid getting banned for posting in this thread I will now make a cockjoke:
Roosters are also known as cocks. Cock is a synonym for penis. Hahaha.