Similar in Britain; I remember seeing several campaigns to raise
money so children who couldn't get treatment there could fly to
America for it. The only way that the British government can
continue to fund the NHS is by keeping wages very low compared to other
countries (which is why British nurses often go to work for private
hospitals in America), restricting use by older people (and lately
they've been limiting access to some treatments for smokers), running
down the infrastructure and increasing waiting time for operations.
In Britain, if you're over 70 and need a non-essential operation,
odds are high that you won't get it from the NHS; if you do, you'll
probably have to wait a year or two. How many Americans would put up
with that kind of treatment, when with private healthcare they could
have the operation in a few weeks, if not tomorrow?
Of course a very large part of the cost of health-care is set by the
licensing restrictions which limit the availability of medical staff.
I don't know how those compare between the countries in question. A
further issue is how long you try to keep people alive; I strongly
suspect that US private hospitals would continue life-support well
beyond the point when the NHS would pull the plug. That isn't cheap.