That is not the reason. Billing in the USA is completely foo-bar'd but part of the underlying problem is the large fraction of unpaid and under compensated services. Others make up the difference. E.g., Medicare pays ~ $10 for a doctor visit.
As an aside, while it is certainly true that Americans pay a world leading fraction of their GDP for healthcare and are not healthier overall, the reasons are complicated and varied. Part of the story is that Americans are a fat, substance abusing, hypertensive, diabetic society. Another part of the story is last year of life costs. And another part of the story is a demand for the latest and greatest, all the time, now. And not to be forgotten is the direct and insidious costs that come with living in a hyper-litigious society.
A single payer system does not solve any of these cost drivers.