To be clear, I misspoke in my closing sentence, and intended to say "It seems likely to me that the political reality in America would prohibit banning private insurance" -- but got mixed up in my negatives and now it's too late to edit.

That said, answering your question, they would presumably not ban it because a) the aforementioned arguments upthread make it pretty silly to do, and because b) it seems impossible to me that banning a wholly legal service just because it competes with the government would ever be deemed as constitutional.

Well, not ever -- but not today, and not with the direction that the Supreme Court has been taking for decades now in regards to government power. Licensure seems possible, but I think even prohibitive licensure intended to dissuade private entities from competing with the government would be tossed out, and we already have such programs for insurance providers anyway.