You don't really need to ban private coverage in a single-payer plan, but you do need to enroll everyone for best results, so what you can't do is to allow opt-outs.
The differentiating point regarding to the US is really the need to consolidate the risk pool across the population for economic efficiency. That, and using the state's purchasing power to negotiate better pricing, as is in common in other fields of endeavor in free markets. The rest are details.
The reasons you need to enroll everyone are:
to avoid ending up with a pool of unhealthy people while everyone else with better prospects are covered by private plans or opt out, trusting for example that they are young and healthy. This pooling of risk is what makes universal coverage in places like Europe and Canada come in at about half the GDP slice of the US's private + public (Medicare, Medicaid) outlays.
to avoid the excessive paperwork and verification that seem so characteristic of the US system. If everyone is covered by the same plan, then there is no need to check that they are not free-riding by hiding, gosh, being ill.
At that point, you can choose to allow or ban private coverage, including for fully duplicate care.
Canada bans it, by principle that people's access to health care should be equal regardless of their wealth. In practice, lots of union, government and military folk have tailored health plans that don't put them in the waiting lists of the larger public. And a lot of richer folk do health tourism, often in the US.
France is more relaxed. Sure, you can totally pay out of pocket for privately-funded care, but why would you want to?
In both cases, the base national plan is really pretty good, though someone with a really large amount of money could purchase better service in a top US health facility. There are long waiting lists in Canada, but mostly not for life-threatening issues, although there may be waiting lists for diagnosis, including of dangerous problems.
Canada's wait lists are not structural to its approach - given more funding, they could be reduced. But, at some point, you run the risk of health care funding crowding out all other government spending, including education. That is not a desirable outcome. Perhaps money could be redirected from end-of-life spending, where supportive care is more realistic than curing someone, but that is a thorny ethical issue. France's plan is more generously funded and usually figures heavily in national budget deficits.
Note that in either case there is no strong incentives for insurers to sell coverage that duplicates what the single payer system is already offering. Canada's position is more a matter of principle than practicality.
Both countries totally allow for private providers - doctors and hospitals need not be government employees and funding is also different - provincial level/general tax revenue (+ individual topup) in Canada, national, with complicated funding plans based on employee status in France.
Private insurers do exist, and make substantial money, offering top up coverage in areas like dentistry.
On the other hand, the US health sector sucks up nearly 18% of GDP vs 10-12% in European systems (rough numbers). That much pork employs a lot of people, pays a lot of stockholders and greases a lot of politicians via lobbyists. Plus, people in generous existing plans, and that includes US retirees who magically turn pro-single-payer at 65, don't want to mix their lot in with the hoi polloi.
This is the flaw with Obamacare, though well-intentioned, it doesn't go far enough because it couldn't get the votes to do it.
Don't expect US single payer, with or without allowance for private coverage, to pass anytime soon. Too much money to fight against it and too common for Americans to paint it with the term socialism even when their government's share ends up spending as much of its GDP as everyone else (Medicate, Medicaid), before the private sector pays its pound of flesh.