There will always be a private market because there will always be doctors who will offer services if you pay them enough.
Healthy people end up paying for the unhealthy. There is no magic single payer with unlimited resources. Resources that could have funded other things that society needs are instead spent on healthcare. And ultimately, since resources are limited, some decisions will need to be made about what is worth saving -- and it won't depend on how much someone has contributed to society, it will only depend on expected outcome -- how much money for what kind of prognosis / most likely outcome. In such a system, more government money would be invested into reviving a 31 year old drug addict who overdosed on heroin than a 81 year old scientist who cured a rare disease, but now has skin cancer.
With a secondary private insurance market, you continue to have the capitalist model and freedom of choice.
That's the core of the issue: how do you ration a fundamentally limited resource. I disagree with your belief that our health care market is based on how much someone has contributed. Your hypothetical example simplifies the issue to an absurdity. In your example: the 31 year old drug addict could be the son of a successful personal injury lawyer so he's got his cure. The 81 year old scientist could have been dumped by the insurance company because he's old and thus a burden on their profit margins.