I don't know where this myth comes from -- is it just a misunderstanding of how public health care works, assuming that it's an infinite money pit?

Once you buy into that, you start applying that faulty reasoning to the real world, and you come up with afactual claims like the original poster did.

Perversely, it also raises costs by increasing demand for medical care.

source: i work in healthcare.

This is true for Medicare and private insurance. Think about it. You buy a car based on how much you can afford. If I offered to buy you any car you wanted regardless of price, you would pay much more for a car than if you yourself had to pay. If this were true for everyone, auto dealers would no longer have to compete on price, but would instead would compete on giving the best service, raising costs even further.

Historically, health care costs began to greatly rise when the federal government made it a tax deduction for employee health insurance. This pushed everyone into private insurance. Costs have spiraled ever since. The solution (as proposed by both candidate McCain and candidate Romney) is to shift the tax deduction to the individual for medical costs. This allows people to afford health care, but also introduces competitive market forces (from a first-party payer) into the equation.

Except that's not what Medicare does. You may not have an incentive to get the cheapest treatment, but the people running Medicare do have an incentive to keep costs down, and the power of law to do so.