There are a few elements to this.
Many providers, including public providers, are exceptionally reluctant to see uninsured patients. If push comes to shove they cannot outright refuse, but they can be highly disinclined.
Insurance isn't just "we cover your bills", but "we negotiate rates with providers". Much healthcare billing is basically funny money. Or more specifically, there are huge fixed costs, but charges are based on specific services billed (HMOs excepted, though internally they likely operate somewhat similarly), so that what in a tremendous number of cases, the healthcare provider cannot tell you, as a private individual, what you'd be billed for treatment. But some hospital procedure or stay can billed entirely differently depending on who paying, with hugely varying rates.
And, since the uninsured has zero effective bargaining leverage (health plans can threaten to drop a provider or apply other leverage), they get the highest rates.
On top of all this, billing is tremendously disaggregated. You might see charges for facility, individual supplies, labs, transport, and individual medical personnel who charge independently. Many (if not most) doctors aren't employees of a hospital, but are independent service providers who charge separately.
It's a mess.
And a distressingly ineffective one. There's been very little real improvement in health outcomes since the 1950s (and for quite some time they got worse). Much of the improvement can be chalked up to improved environmental and lifestyle factors (removing lead from paint and gasoline, asbestos, less smoking and drinking). It's been improved health outcomes for the poor and minorities which account for much of the life expectancy gains as well. Earlier improvements largely came from public health measures: fresh, clean drinking water, sewerage systems, municipal waste removal, and quarantines for infectious disease. Even vaccinations and antibiotics were relatively late and minor interventions by comparison.