I don't understand why some people think it comes down to a binary choice of the NHS or the U.S. system.
If the NHS was so wonderful, then the entire planet would have followed that model. They haven't, so what does that tell you?
Many countries in Europe have a co pay system, which works well. Where I live, physios, radiographers, radiologists, cardiologists are in clinics, or their own practices, away from the hospitals, and you are referred by your GP and can make your own appointments.
We need a grown up conversation about the structure of the NHS; what it does, how it can be more efficient, and how we fund it. Until that is done, and a system emerges that is fit for purpose, we are going nowhere with it.