What you're neglecting is that in a market where there is free competition, the consumer is free to penalize bad services by simply going somewhere else, whereas there is an incentive to provide good care. In a government monopoly, there is no incentive whatsoever since the patient has no alternatives, the only incentive is to perform a bad service so there will be problems and funding will be increased.
Only thing is this: what of the people who couldn't afford what the private providers were charging? While the free market might provide competition, one thing that it doesn't do, and was never intended to do, is to provide goods or services to those without adequate money for them. The free market might maximize financial utility of consumers and producers, but that's as far as it goes. If a company's main goal is to turn a profit, which is the goal of all private for-profit companies that sell a product, what possible incentive would they have to provide their service to everyone?
From just about everything I've seen, people's arguments in favor of a public health care system have less to do with how well the system runs and have more to do with the idea that health care is a right that everyone in the entire country should have access to, whether or not they're capable of paying what a private company would charge. As such, what you're arguing is largely irrelevant, even if it's true.
