Meaningful competition cannot exist between tax-funded and voluntarily-funded service providers. Anyone who chooses the private provider ends up paying for both. The only way to ensure a level playing field is for the "government" entity to be limited to essentially a co-op, receiving no government funding or special influence and adhering to all the same rules which apply to private providers, in which case it might as well not be a government entity in the first place. Even then there is an obvious conflict of interest since the organization that makes and enforces the rules is also backing one of the competitors.

Health care was way less bureaucratic in Italy. Every time you go to the doctor in the US, you fill out forms. And then hand them to the person in the office who spends their days doing insurance stuff.

In Italy - you just go to the doctor.