Not really. Private health insurance and fully private hospital care exists in most (all?) states with full state-sponsored medical coverage. Ultimately, if people don't have to pay for health insurance any more, a huge proportion won't. Which means the state becomes the single largest buyer of healthcare, probably bigger than all the other players put together, and so should be able to get better value for healthcare than the private companies ever did. Better cheap treatments than a half-empty hospital. And a state health insurer also doesn't need to advertise itself to consumers or pay shareholders, so it should come out ahead even if it's unable to pay any less per treatment than the private companies.
Of course, it's still dire for the health insurers because their customer base shrinks massively when private health insurance becomes a luxury to get people more expensive and faster specialist treatment as opposed to the only way to be able to afford to not die from expensively treatable ailments. But there's no reason to assume that some/most wouldn't continue to provide the option of more expensive and more exclusive healthcare for those that don't trust the publicly funded system or want to beat the queue.