Bottom line. The more providers competing in the market, the less coverage for the population as a whole, the less claims paid, the more profit for our special people, those forever wonderful capitalist pig hegemons. This is how neoliberalism works.
This general principle of generating many optional systems also works very well as a propaganda tool. For any given moment where univerbal healthcare reaches some pivotal turning point, it is always a good idea to generate as many slightly different plans as possible, so as to flood the world of ideas with utterly meaningless variants. Collectively these will muddy the waters of insight and divide the myriad of interests who have temporarily formed a coalition. Dissolve the coalition, and you dissolve any coherent movement.
[WS:] Actually, what you describe here is the effect of not the free market per se, but of the sharp class division of the US society. Fundamentally, you have the class of haves - those who can afford expensive and overpriced services (that works for health care, as well as for housing, transportation, restaurants, etc.) and the class of have-nots - those who cannot. The have-nots however are not totally excluded - they have some access to health services either paid for by government or mandated by law or government regulations.
The problem is, however, that the providers - be it doctors, hospitals, landlords, transit companies or restaurateurs - do not make as much money by providing the services for have-nots, because of the fixed fee schedule for such services. Given the limited capacity of any individual provider to deliver services, he (or she) acts rationally by diverting all that capacity for serving the haves (i.e. where the profits are) and by implication avoid serving the have-nots, where the profit making potential is not that great.
The net effect of that situation is not the shortage of services, as you claim, but the glut of services for the haves and the shortage of services for have-nots. This can be observed not just in the health care, but other services industries, especially housing (the glut of expensive housing amidst the dearth of affordable housing.)
If this analysis is correct, a single payer system (a monopsony) will not do much to correct the problem, since the providers will still be chasing after the wealthy patrons at the expense of those who rely mainly on public health insurance. The only way to address it is really elimination of class inequality or at least achieving some form of equalization in the access to services.
That equalization can be better achieved, in my view, on the supply side rather than the demand side. That is, instead of subsidizing the cost of service delivered by private providers, the public sector can compete with those providers by setting up publicly or cooperatively owned service centers (clinics, hospitals) serving anyone at a reasonable cost (comparable to "customary and reasonable fees" set forth by insurance companies.) If such centered were running operating loss, the loss would be subsidized from public funds (the way the airlines or defense contractors are.) The key is some form of public ownership of such centers, which would guarantee their compliance with their statutory mission (i.e. service at or below the cost.)
A similar situation has already developed in the retail industry, where we have a two tier service system - expensive boutiques serving the haves, and the wal-martized "big boxes" serving the have-nots. Whatever one may think of the morality or "political correctness" of that solution, one cannot validly claim that the have-nots face a shortage of consumer goods (of comparable quality but different packaging!) in this country. And that is the goal here - providing the needed services to those who are not in a position to pay a premium price for them.
Wojtek