But it would be paid for by taxes spread across the tax-paying citizens. It would appear to all people in the US that their healthcare is free as they are not directly paying for it. Hence infinite demand.
This isn't a free sandwich promotion that ends with irrational people waiting hours in line! People are not going to go hangout in doctors offices, hospitals, clinics, etc... for fun. Most people that have good insurance, like myself, almost never go to the doctor because it's a pain, not because of a co-pay.
And if by some chance there is a big increase the use of medical care / services ... maybe that's a good thing, that people are going and finding issues before they become severe life threatening issues and everyone saves more overall in the long run. And they have better quality of life and less suffering from going untreated... Did you consider that?