assuming you're asking in good faith, the answer is pretty simple. A larger population has more leverage, and if all of the citizens bargain as one bloc (i.e., if the government does so on their behalf), it drives the market. Note that a tremendous chunk of the market needs to be involved for this to have maximum effect -- if a physician can "opt out" of Medicare/MediCal and make more money, they will, by and large. (There's a reason that being on call instead of waiting for referrals is called "service")
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.