It's a common claim that health care would be more efficient and cheaper if not for third party payment. Sometimes, yes, but often these claims are overstated, especially when the link between treatment and improvement is murky.
To consider one example, for the most part autism-related services are not covered by private health insurance. Government aid is often scarce as well. Also in Canada medical benefits for autism-related services are quite limited. So when it comes to autism, this is a fee for service setting for the most part.
And what does this world look like?
1. Services are not especially cheap nor do they seem to be falling in price.
2. Market participants are not well informed about what works. Many parents of autistic children pursue hopeless treatments or unvalidated or even refuted theories. Some of the treatments, such as chelation, are harmful in many cases and yield no benefits.
3. There is lots of innovation — in terms of advertised methods of treatment — but it is unclear, to say the least, what percentage of these innovations succeeds. Very often it is parents "buying hope."
The point is not that insurance coverage would solve all these problems. Third party coverage would slant the relative prices toward more mainstream treatments and away from the fads; how good or bad this would be depends on your point of view as to what brings better (worse) outcomes.
Overall I don't view the autism example as a good selling point for the view that third party payment is the basic problem behind U.S. health care. Nor do I see critics of third party payment citing autism services as a model example for their ideas. (By the way, it is an open question how much autism should be an education issue and how much it should be a health care issue; de facto it is often a health care issue but this should not be taken for granted.)
Another lesson is this: the more emotional the issue, the less effective any health care system will be. Policy discussions of "health care" often require more disaggregation.
Addendum: There is, by the way, a movement afoot to require that private insurance cover some autism-related services, such as ABA. Given the costs of the treatment, and the unclear link between treatment and results, I would be curious to hear if "universal coverage" advocates would include this in their ideal public policy. I would say they should admit that any notion of "universal coverage" is value-laden rather than purely descriptive.