Actually there would be a way back.
Most countries that have public insurance also have private insurance. Just because the public sector is there it doesn't mean that the private sector ceases to exist. It does however, force them to operate more honestly since if their service is not reliable, over things like health care, if given the choice, people will choose reliability.
It's like saying the economy class seats on an airline will make first class and business class seats a thing of the past. Not true.
Ideally there would be a public version of just about every service or industry but this would not be sustainable since public run programs are inherently bad at turning out a profit and do tend to be wasteful. So it's best to simply save these for the most important industries (school, healthcare, what have you). Public option does not mean the private sector will go out of business though it does mean the less competitive ones will.
Even if there is a free cafeteria for all, most people will still go to the Italian restaurant, the Irish pub, the Chinese takeout etc. But let's say you've been really out of luck, you've lost everything. You can still get food. You won't have much of a choice but you'll still be able to get it.
Now that's just an example. I don't advocate nation wide tax run cafeterias since I haven't exactly put a lot of thought into that specific idea. I'm just trying to show you how a nation wide, tax run entity doesn't automatically put the private sector out of business.
The only thing that can really end the private sector will be to illegalize it, and that's when you start stepping into the territory of Communism.