I live in a single payer system country and it's nothing like what you describe.
First, how it is paid. It's text book socialism. People pay into it based on ability and receive benefits based on need. It doesn't run at a loss.
Second, the level of care. Basically the government puts a lower bound on health care and uses taxation to do it. You're guaranteed a basic level of care regardless of your situation. You can opt for private insurance that stacks on top of that. A basic level of treatment is covered by the government, but anything experimental or exorbitant you will likely have to pay yourself, unless you have the aforementioned gold-plated insurance. Insurance is much cheaper because it only needs to cover unusual care, but of course I do pay a lot of taxes.
It's not different from a private system, because you can't receive unlimited care, but unlike a private system you're always guaranteed a minimum care.