It's interesting how you mention the French Plan. If I may, allow me to describe the way it is done in Israel. There has been a system of HMO's that began in Mandatory days by the Social-Political organization of the Yishuv, the Jews of the Mandate. You paid a subscriber fee, and in turn you received the medical services. By the time the State was established there was an infrastructure that included clinics and hospitals that served the population of the Mandate. Once the State was established, the "political" side was loath to give up their regular income, and continued to maintain the HMOs , and received government subsidies to assist it. You were a member of on the "funds", and each fund had its own rules.
To move from one to the other was never easy, but there were those that did. Indeed, membership in one, or the other was one of ways the Labor Party kept people "loyal", after a fashion, and continued to be in power for over thirty years. It was one of those systems, where someone that was a member of "the other HMO" knew that his turn in the line in the Employment Bureau would not net him the "better positions", and in some cases, kept them form getting promotions at the workplace. Gradually, the Histadrut, the General Labor Union, revealed "leaks", its banks were found to be less than honest, subscriber contributions were used to promote the Labor Party during elections and other "interesting" features. It came to a head when one of leading figures in Bank HaPoalim, the financial institution that kept the Labor Party afloat committed suicide after he was alleged to have embezzled funds.
The fate to the "independent" aka government subsidized HMOs was sealed when the Likud, the opposition finally changed the system. All the HMO's were fully supported by the government by a taxation. You could move from one plan to the other without any penalties. It did result in some surcharges and co-pays that didn't exist previously, but hardly anything that was costly. In addition, it opened the possibility for the HMOs to proved Alternative Medical Services, such Acupuncture, Homeopathy, Naturopathy, and other forms of Complimentary Medical Services, Dental services, too, that were never covered by any HMO, ever.
Literally, all the medical services, Conventional, and Complimentary were covered by all the HMOs. Except for the very specific and special services, most medical costs are quite reasonable. I say, that I had both cataracts removed in a local hospital by the best specialist in the region at the cost, in dollars $US 25. I was treated for a heart procedure, for . . . $US 5. The other day, I was called in for an eye examination that I had been neglecting for a number of years, which cost me nothing at all. Granted the Complimentary Medical Services to cost something, but hardly as expensive as the same treatments would cost in the States--figure that a package of treatments that include Acupuncture, Tui Na (a Chinese from of Medical Massage), Naturopathy, Homeopathy, twenty or so in a package that costs $US 300, and the payments can be spread out over six months.
National Health in the States is possible, only if people stopped blathering away about "Socialized Medicine", and sat down and looking at the facts before them.