Government provision of health insurance means that every procedure covered is a political issue (e.g. abortion for the left, sexual-orientation counselling for the right). This means that there will be nationwide winners and losers, and no-one but the rich will be able to simply change providers, as with a private market. As an example, look at how public schools are run: only the rich are able to opt out of government schooling.
American governments are not capable of running efficient programmes, because to a first approximation American government employment is a make-work program for people who can't get hired at anywhere that cares about competence. That means that a government insurance provider will be incompetent and inefficient.
To make things worse, the employees of said provider will be unionised, and that union will do its best to make it even more inefficient (e.g. take a look at the prison guards' unions).
American governments are not capable of running efficient programs because to a zeroth approximation Americans keep electing politicians that not only promise that government can't do anything right, but also work to ensure it.