What annoys me as an American about our health care is not that some other countries achieve better outcomes at much lower cost--it is that they do so in so many different ways.
For instance, if they all did it by having the government completely run the health care system, with all doctors working for government hospitals and clinics, and being trained in government medical schools, with all costs set by the government, then I could understand the US rejecting that as being too much government intrusion. (Note I didn't say I'd necessarily agree with such rejection--good health care arguably is more important than limited government power--I'm just saying I could understand such rejection).
But looking around, what stands out is how diverse the approaches of other countries are. There are some that do have the government finance and provide health care, much the way the police or military are provided by the government.
There are some that use private insurers paying, with care provided by private hospitals and doctors, but the insurance companies are non-profit and regulated.
There are some that provide care via private hospitals and doctors, but have a government run insurance program to pay for it.
There was a good look at how it is done in 5 capitalist democracies here: http://www.pbs.org/wgbh/pages/frontline/sickaroundtheworld/c...
See also the link on that page to the explanation of the four basic models used around the world. That page includes a striking way to explain the other systems to Americans:
These four models should be fairly easy for Americans to
understand because we have elements of all of them in our
fragmented national health care apparatus. When it comes
to treating veterans, we're Britain or Cuba. For Americans
over the age of 65 on Medicare, we're Canada. For working
Americans who get insurance on the job, we're Germany.
For the 15 percent of the population who have no health
insurance, the United States is Cambodia or Burkina Faso
or rural India, with access to a doctor available if you
can pay the bill out-of-pocket at the time of treatment
or if you're sick enough to be admitted to the emergency
ward at the public hospital.