I was on Green Mountain Care https://en.wikipedia.org/wiki/Vermont_health_care_reform for a year or two when it was available in Vermont... it was absolutely genuine single payer insurance system that gave me access to no-copay services. Sadly the program was shuttered three years after it had been introduced but I remember it fondly. It was accessible, affordable, and extremely comforting to have access to.
Yeah, I think this is the right answer. It's a bit shocking that none exist, but that doesn't mean it's untrue. Some may have existed but don't anymore (like the Vermont example in another reply).
The health insurance racket in the US is strong -- probably the strongest in the world -- and they have to maintain constantly expand their profits.