I'm still thinking about this topic and I will refer you to a response by Frank Pasquale. He knows a lot about the topic but I'm still not sure how to translate his points into econspeak.
Here is another way of looking at my original question: some of the key (possible) problems in private insurance markets are adverse selection and lack of transparency in prices and terms of service. Previously I asked whether the public plan does compete with the private plans for the same pools of customers. Today I am asking whether greater competition necessarily improves these outcomes. (It's trickiest when it comes to transparency; you might think that a more transparent plan will outcompete a less transparent plan, but the whole point of the lesser transparency is to make the plan look more attractive than it really is. A truly transparent plan could look not so attractive at all. The model here has many relevant iterations, with unclear results. You get the best results when the presence of a transparent plan educates consumers about the nature of the market more generally, rather than just educating consumers about the nature of the transparent plan. But that is hard to do)
As Pasquale points out, a lot of state insurance markets are currently not so competitive. As it stands today, do the more competitive markets offer superior performance? I genuinely do not know but I would like to see the evidence on this question.