"So if the government isn't running health care, who is?"
This question was submitted by a health policy expert. I laugh when people separate the "government" from it's citizens because, essentially, we are the government. Every person elected and appointed in our government system went through a growing-up process similar to everyone else and ate McDonalds, and dated, and got pissed at standing in a post office line. Granted, most politicians grew up richer than us, but that's another post for another day.
In America, we hate having someone tell us what to do. So when the ACA was approved, though it helped lots of folks, people hit the roof because they didn't like that "government" (i.e. "others") was telling them how to make their own health care decisions.
The government serves amazingly innumerable functions. Defense, safety, water, transportation – the list goes on and on. But we've also seen sheer ridiculousness in execution: the Postal Service, mortgage applications, Congress getting paid to not do their jobs, etc.
So I would assert that having P.S.Y.C.H. be run by state-specific nonprofit corporations is a good mix between the government/public and private. Nonprofit companies are subject to specific governmental regulations; thus the government can have the ultimate say in compliance without having to actually run the logistics. The benefits of avoiding governmental control are numerous.
It's hard to fire public employees who aren't doing their jobs. But as a nonprofit, employees requiring remediation can get it, and if things aren't going well, be released to pursue other jobs.
It's also hard to reward public employees who are innovative and efficient because of rules related to seniority and tenure, etc. As a nonprofit, the board and CEO can more easily offer incentives for innovation and efficiency and be much more flexible in meeting the needs of its customers than the government could ever be.
All healthcare providers and places would essentially become contractors and be paid a standardized fee that would be determined yearly. Since people won't have their health care tied to employment any longer, no one would have to stay at a job they dislike.