Well over $570 billion dollars per year are wasted on administrative overhead when firms have to advertise and compete with one another while a further $350 billion is fronted to the medicare system when sick, uninsured individuals are treated in ER, as opposed to receiving preventive health care or merely seeing a family physician. This is why US healthcare costs are 16% of their GDP to cover about a third of the population while the UK pays about 7% of their GDP to healthcare while providing coverage to everyone.
More importantly, services, information, and options are within one singular system which greatly increases efficiency in terms of the administration and application of medicine. As opposed to someone needed multiple plans and separately employed doctors/specialists/hospitals to cover only a small portion of different health services, all of this is contained within one unified system which allows equal access to all of them. Hence why the US was ranked 47th in healthcare efficiency, while Sweden was #10.
Finally, there is far more funding available to important prescriptions and medical equipment as a single-payer system has a consistent funding model that doesn't require an allocation of funds towards advertising. That and preventive healthcare is included, meaning many illnesses can be alleviated far before you would even qualify for coverage in the US. As well, the existence of only one provider, or price controls on costs, ensure than said provider only focuses on providing the best health coverage they possibly can as opposed to as much profit as possible: private providers are focused on making more money than their competitors, that can often preclude modernizing or advancing their service if it would cost them too much money. And being more competitive often means eliminating services and using cost-cutting measures which negatively impact how much funding is available to things like medical equipment or hospital care. Having a funding model that ensures a constant supply of funds to important medical tech and equipment means that the system is always in a position to innovate FOR THE BENEFIT OF THE PATIENT, while still ensuring that coverage is universal and expansive as opposed to selective and limited. Multiple private competitors=multiple firms competing to be as profitable as possible while limiting the amount of funds available to healthcare professionals and treatment options. Again, this is why you're much more likely to survive a life-threatening procedure in other countries as opposed to the US, unless you are extremely wealthy and live in a wealthy region.
Also, demand for health-insurance is by its nature, high. High demand means that under market conditions with no price controls, or no government control, prices will be naturally. While at the same time, private insurers have no reason to keep prices low as there is no threat of customer migration.
Long story short, it shouldn't even be a debate, you just need to look at the evidence.
For one thing, there is no such thing as free health care. Even if there is no direct costs to patients, taxpayers are forced to foot the bill for medical coverage. Moreover, without adequate market forces at play, people will be less likely to become doctors, causing a shortage of health care services and therefore lowering the quality of medical care.
No, universal health care does not provide superior care, because there is no incentive to provide efficient service or to innovate. Knowing that the consumer only has one option, the provider has no reason to try harder to provide a better service to the recipient of the services. Anyone who has sat in line at the Department of Motor Vehicles knows how slow they are. Similarly, the postal service is very slow compared to private services. Competition makes providers compete harder for quality service.