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MeLooking Your first sentence sort of answers itself. It says many Americans cannot afford health care. Your second sentence points to why this is so hard to solve.
Looked at from one of the many countries of all political flavours that do have national health and wlefare systems, I see many Americans are desperately afraid of anything seeming even slightly "socialist".
Even though such welfare systems are simply a nation looking after all its citizens, especially those unable to afford hospital bills, medical insurance or even quite modest personal-pension savings.
No-one denies that a proper health-care system is expensive, even for a nation as wealthy as the USA seems to be; and has to be funded somehow. Either you fund medical treatment via direct fees and personal premiums to insurance companies, or at least partly from national funds needing tax contributions.
So you all pay in the end, but the private, commercial way involves a profit element not going to the hospitals, before anyone waves a stethoscope over you.
The example I know and indeed benefit from personally, the UK's National Health Service, works by Treasury (tax-paid) funding generally, but we still pay for some routine-care services.
Those are provided by direct companies, not insurers, with their NHS-registered patients paying part of the treatment, the NHS the rest. These are dentists', opticians' and some audiologists' fees, dentures, spectacles and hearing-aids; and a small part of the cost of prescription-medicines.
All hospital and General-Practioner services are free to the patient at point of treatment; but of course we all contribute to those via our direct and indirect taxes; and the NHS does use a lot of contractors.
I think President Obama's policy would have been a bit like that, albeit still with private health insurance.
We can of course buy private treatment but it is expensive, usually paid for via insurance. A few major employers include health-insurance help in their salary and pension agreements, but that does not help everyone! The invoice includes the insurer's and providers' profit shares, and the policy no doubt excludes various illnesses or injuries not commercially safe. (Just look at any holiday insurance, for example.)
The NHS was inspired by co-operative mutual-assistance savings-plans established in the mid-20C among the poorly-paid miners' families in Labour MP Aneurin Bevan's constituency in South Wales.
I see the USA's two-party political spectrum as generally somewhere Right of most European nations', but even those countries are far from the "socialism" evoked by Cold War and MacArthy memories.
Indeed, some Americans who can presumably afford it, propose a complete sink-or-swim ideology, judging by certain hard-line manifestos and informal comments I have read. (Their own words, not what someone else wants me to think they say!)
So to answer your question, you had a sort of half-way house in which the State pays from tax-revenue for at least part of the treatment for those unable to afford commercial insurance and treatment fees.
The United States of America could create and afford even afford a full system modelled on many other nations', but that's not the question.
The question is whether her citizens want it.
In the end their choice is purely by political ideology and election results.