I suspect, but don't know for sure, that private insurance can have no long term role in health care. The only role they seem to serve is to be a middleman who takes a substantial cut in return for work that is really pretty trivial, and insulates providers from consumers in a way that prevents the free market from working.
Whether medical care should be completely nationalized or can be implemented by a mix of public and private hospitals and other providers isn't clear to me, but what is clear is that the present system is rife with gouging and prevents a sizable part of the population from getting health care.
Here's my strategy:
1: Fix the PPACA. The mandate is critical. The regulatory constraints on what counts as acceptable insurance are critical. There are lots of details that need to be polished. The price of drugs needs to be brought under control. The numerous areas where there is huge gouging need to be regulated. (e.g. MRI) The medicaid expansion needs to be mandatory for all states and everybody needs to be eligible. If you have a contagious disease, you need to seek treatment, whether you're legally here or not. If you're afraid to go to the hospital because you might get deported, and you're sick, you are spreading disease.
2: A public option. The insurance companis exist to make a profit, and in far too many cases, they're making the bulk of their money by gouging. In addition, they are a major insulator between the provider and consumer, totally preventing any semblance of an effective free market. A public option would be an insurance company with the resources of the government behind it. When this has been tried before, it has been vastly more efficient than private companies. I am pretty sure that the insurance companies will not be able to compete with a public option, but I'd be more than happy to be wrong about this. In that case, the public option would simply be another option, which could help out in places where insurance doesn't reach for some reason. But I'm pretty sure that won't happen, and the insurance companies will fight hard and dirty, by lobbying, bribery and outright cheating, to keep their perquisite.
3: After a few years of a public option, most of the insurance companies will be gone. At this point, there will be three systems: 1: the various existing government systems, such as the VA and medicare. 2: the various existing employer based systems. Most of the insurance based ones will have already been turned into versions of the public option by this point, but the self-insurance ones may still exist. 3: public option. The employers and many employees will be seeking a way to rationalize this. I suspect the best way to do it is Medicare for All: Simply pay medical costs like any other government service. This is by far the cheapest way to do it.
None of this would ban private insurance. If you want private insurance over and above medicare for some reason, you can do that. More realistically, there are plenty of things the government shouldn't be paying for, such as optional cosmetic surgery, that will be paid for out of pocket, and that's fine. Other concierge services would also be perfectly ok.
I'm ok with a nominal cost per visit. One of the things that happens in government systems sometimes is that hypochondriacs and bored people often spend a lot of time with the doctor just to have someone to talk to. To some degree, this is ok--hypochondria is sometimes a cover for some more serious underlying condition. So charge a few dollars per visit, just to keep abuse to a minimum. The provider may waive this fee at the behest of the doctor.