Do you think an entire industry is going to let a dysfunctional, bureaucratic government to dictate what their rates are? There will always be a market for better care to the highest bidder.
Way to load your question, as if the entire US health insurance industry is not already dysfunctional and bureaucratic ...
But to answer your question, sure, there will be always a market for the select few for whom money isn't a concern. The difference is that, under a single-payer system, you don't have to worry about it: just find a doctor at Yelp, or let the ambulance drive to the nearest hospital, and your doctor and your hospital is covered. You will intentionally have to go out of your way to find your boutique doctor who doesn't accept "dysfunctional government insurance."
This is very important, because when your kid is bleeding inside an ambulance your concern shouldn't be "Wait but does this hospital accept my insurance?" You have a bigger problem to worry about.
