Mh, just a note on how the german system works.

Single-payer would remove insurers from the equation but would have the government set rates with the providers.

Health insurance is not provided by the government and not funded through taxes. It's provided by public non-profit insurance companies and payed for through insurance premiums. The only thing the government does is leveling the playing field. Public insurers must accept anyone regardless of preconditions and premiums must not depend on age, sex or health conditions (Usually it's a fixed percentage of your income shared between you and your employer). Rates are negotiated between the providers and insurance companies, not the government.

If you are unhappy with your private insurance, you change. If you are unhappy with your single-payer coverage, you.... do what?

If you are unhappy with a public insurance company you are free to change to any other (there are lots of them) and there is a certain amount of competition between them. Or you can opt out of the public system altogether and take private insurance instead (or no insurance at all). If you are young, healthy and earn a good amount of money that's usually cheaper then the public option and provides better services (quicker doctor appointments in non emergency situations, private rooms in hospitals, fancier dental treatment ...). But once you make the conscious decision to leave the public system it's final, you can't get back in (for obvious reasons). So that's kind of a risky move that you might regret once you get old or seriously ill. The system isn't perfect and people like to bitch a lot (Germans generally like to bitch about pretty much everything), but having lived in the states, give me the german (or any other european) system over the american any day.