Since you edited your comment but didn't reply to mine, I'll respond to your second part.

If you're genuinely wondering how you get psychiatrists to accept public healthcare, then it's improving wages and working conditions for psychiatrists who work at Health + Hospitals. Psychiatrists are already part of The City's "public healthcare" system through Health + Hospitals. And in the shorter term, it's expanding the number of openings for psychiatrists and therapists.

In any case, having nothing but criticisms helps discuss the issues with a plan, what helps even more, especially if you are a physician is providing your own solutions to the issue.