I think a two tier system makes sense. Public clinics provide basic services and emergency treatment at low/no out of pocket cost for anyone who walks in the door. Private providers are free to offer any services they choose. Seems like a win-win. A baseline of care exists for everyone, but people are free to seek services elsewhere if they prefer.
This is basically how schools work already. Public school provides a basic education, but you're free to send your kid to private school if you have specific desires that aren't met by the public school.
Before you point out that public schools don't do a great job in a lot of places, there is one key difference between schools and healthcare facilities. Outside if emergency situations, patients have the ability to "vote with their feet." If a clinic in the Bronx is offering substandard care, you can hop on the 2 train and head to Manhattan. If you tie compensation to patient volume, there are economic incentives to offering high quality care.