You keep inferring that a single payer system would reduce choice when the reality is that the opposite is true. Instead of a private insurer assigning you providers based on who is in their network, every provider would be in network. In the vast majority of scenarios, the patient is two degrees separated from true freedom of choice, as options are limited by insurer networks and insurers are limited by the patient's employer. If you are concerned about a government monopoly, I would suggest that private insurers be able to compete with a single payer system in the same way that private schools compete with public schools. Removing the insurer's financial incentive and imposing a statutory obligation to cover care means that the theoretical single payer system is not picking losers in the way that a private insurer currently is.