That's true, and it is, but people's inability to be clear-eyed about the politics and the incentives are a big part of why we're locked in this situation. Proponents of payer-side reform are, from the perspective of a huge number of Americans, selling a bill of goods. Ordinary PMC suburban households will very likely be materially worse off.
We have done a number of things about it, most of which are more expensive and less effective than the very simple option of just adding them to Medicare, but expecting people to change from what they know and like to benefit less than 10% of the population is not a winning strategy.
Medicare shares almost the problems of the private system and adds others. The current Medicare scheme, of single-payer health care for the cohort most in need of health services, makes a lot of sense.
Medicare does have many problems and arguably isn't cheaper than our current private systems, but I would say a large portion of the uninsured also are part of the cohort most in need of health services (or they're young and healthy and cheap to insure/treat and largely irrelevant), which is why I think it would make sense to just roll them into some public plan instead of using EMTALA as a catastrophic insurer of last resort.
I agree and am generally sanguine with any service we want to roll out to the Medicaid cohort, but recoil from the idea of standardizing all health care in the US on Medicare, not because I'd be worse off (though, like every PMC household in the US, I probably would), but because Medicare is not good, and we'd basically be enshrining all the failures of our system in law and hiding the costs in the tax code.