First, in response to your earlier comment to me, I was going to point out that the "unfairness" is not a smokescreen to the question the insurance companies are trying to ask, but you're answering a different question (which you've just pointed out).

Second, I think it does matter that there is private insurance, and the reason is for competition. Here, I'll point to the British model, where there is an automatic public option but something like one third of the population also have private insurance (including one set of my grandparents, but not the other). Private insurance should continue to exist because I don't think the public option should be funding all the medical options available (easiest example is types of assisted living, big gradients without necessarily referring to actual health outcomes). Plus, monopoly service stifles innovation (desperately needed in healthcare - wouldn't it be interesting if a private insurer worked out how to cost less than a public option?) and can promote sloppy performance.