We have a test case to look at to see how a single payer system would work: The VA.

How's that working out?

But yes, you could say that Rand Paul's plan is a single payer system: the single payer is the customer, as in nearly every other aspect of our lives.  A customer who cares how much he pays and has to prioritize his spending habits.

We lost control of health care costs in the late 1940's when "group health insurance" was introduced as a way to circumvent government wage controls.  Once we started spending someone else's money for health care, the costs have soared continuously.  It's become a boondoggle for the drug, hospital, and insurance industry because nobody cares what the charges are.

The only areas in health care where the costs are coming down are elective procedures that aren't covered by insurance.  Competition controls prices.