Private health insurance in a country that has a robust public option is an entirely different thing because that baseline level of care means that the private providers have to offer something substantial above and beyond the baseline to justify their continued existence. People with medical needs can choose to tell them all where to stick it and still live without going in to debt.
Most insured individuals in America don't choose their health insurance provider, they just get it from their employer where they might have a choice of plans. The closest thing to an actual competitive insurance market we have is that provided by the ACA, and though that act did many good things the disappointment of the marketplace has been well documented.
Your argument would be stronger if your black box mental model wasn’t so obviously exaggerated.
I live in the UK where we have universal healthcare that is free at the point of need. We also have private healthcare. In most countries in Europe both private and public healthcare exist side by side. Per capita spending on healthcare (across the private and public provision) in the UK is a tiny fraction of what it is in the US and outcomes in terms of quality adjusted life years and measured in individual life expectancy and patient outcomes for given conditions were significantly better than the US last time I checked (which is admittedly a while ago).
The US healthcare system is definitely expensive and delivers a poor outcome, but you’re not convincing when you try to make a pastiche of the system that paints it as purely bad and say that health insurance simply should not exist for pure moral reasons.
