Most physicians in (e.g.) the UK do participate in the single-payer market (there are a small number who make a living offering pay-as-you-go services, but they are the vanishing minority), since it dominates demand. In the US there are a great many physicians who simply won't accept Medicare rates (they're viewed as too low by most) and since there are alternative sources of patients, that's who they treat (typically privately insured). This leads to the cases that show up at County or the ER being a hell of a lot more expensive than necessary as they tend not to be survivable for long. (A running joke at most county hospitals is that conditions believed "incompatible with life" routinely walk or roll into the ER and clinics.)
If you have 1-3 insurance companies and MediCal/Medicaid and Medicare then you have different rates for different groups, almost all of it horrendously opaque, and the 3rd party insurers are not incentivized to pay for anything.
As far as private vs. public, the issue here is the same as for schools, a private insurer or school can choose not to insure or educate a "customer", the government by law cannot. In the handful of cross-over studies of charter schools or vouchers, after controlling for subject-specific effects, the children who switched from public to charter or private tended to do slightly worse than expected based on their test scores from public schools. (It is a difficult experiment to run for numerous reasons.)
Medical care, unlike most goods and services, is stunningly inelastic in demand -- you either need it and will do whatever is required to get it, or you don't and won't, by and large. (Elective surgeries for cosmetic purposes are a separate matter; nobody goes in for a stent "just because" or visits the trauma unit just to poke their head in) Furthermore, a substantial amount of the cost is centered on the first and last few years of a person's life. Unless you would like the "market driven solution" of even higher infant mortality and elderly culling to proceed, 3rd party insurers don't have the incentives to make it go.
Right there is the big problem, though: Medicare reimbursement rates are already below sustainable levels for providers, which actually results in providers charging private insurers for the difference.
If Medicare were expanded to everyone, either Medicare would have to increase its reimbursement rates, or you'd see providers close up their practices (which is already happening, and which is one of the current problems with providing affordable care outside urban areas).
> Medical care, unlike most goods and services, is stunningly inelastic in demand
That's actually not true at all - medical care is highly elastic, as evidenced by the utilization differences for people who have plans with high copays and deductibles compared to those who don't.
> As far as private vs. public, the issue here is the same as for schools, a private insurer or school can choose not to insure or educate a "customer", the government by law cannot. In the handful of cross-over studies of charter schools or vouchers, after controlling for subject-specific effects, the children who switched from public to charter or private tended to do slightly worse than expected based on their test scores from public schools. (It is a difficult experiment to run for numerous reasons.)
But we actually do have a point of comparison here, because Medicare does have both privately managed and publicly-managed plans (as does Medicaid). Consistently, the privately-managed plans come in under budget while delivering superior medical outcome metrics and patient satisfaction scores compared to Original Medicare (or the publicly-administered Medicaid plans).
Lord knows I've had about enough red tape for several lifetimes from NIH, NSF, and similar organs; Medicare as it now stands somehow manages to result in both medicine-at-a-loss and also fraud on a spectacular scale. I'm not a big fan of government but between consolidation and fragmentation, I don't think the current medical care solution is working, nor is it sustainable.
But as long as healthcare providers can find someone else to overcharge then they'll do that rather than accepting Medicare patients.
No, Medicare reimbursement rates are about 7% lower (in the aggregate, not individually) than COGS - the marginal costs of providing care. That is, if a test costs a provider $100 to purchase wholesale, Medicare reimburses $93, which doesn't cover the cost of the supplies, let alone covering overhead (wages for staff, office rent, etc.)
Well, independent private practices have been dying out for precisely this reason, and outside of Critical Access regions (which have a higher fee structure), many don't anymore. Some view it as an act of charity (the way lawyers might take on pro bono cases), but the laws regarding insurance segregation have gotten stricter and the logistics of it have become so burdensome that many just view Medicare as too much of a liability. It's one of the reasons Medicare patients have such horrific wait times to see doctors, particularly specialists.
Hospitals take Medicare because many of them are legally required to. If they're part of larger hospital systems, they can also play interesting tricks to manage the respective patient populations without explicitly discriminating based on insurance provider, which has been another major force behind the massive consolidation of hospital systems that we've seen over the last decade.
The other major force, incidentally, is that many of them have been going bankrupt, and are being bought out either by other hospital groups or by insurance companies.