You have it backwards. Single-payer advocates aren't pointing to a single system, you are. We're comparing the US with every other system in the developed world.
The US health care system is half as efficient as every other system -- every other country is providing an equivalent level of health for half of our cost. So either the US government is twice as bad as every other government in the world -- possible, although unlikely -- or it will do a better job running health care than the cronies and incompetents of the health insurance industry.
By the way, Medicare has the lowest overhead, by far, of any health plan in the country. It's not perfect, but you're still misinformed.