And how many of those "private" insurance companies can still write "major medical" high deductible catastrophic coverage polices that so many of us have used in time past as ... gasp,
insurance, as opposed to a tax advantaged benefit + originally a way to get around WWII wage controls? Etc. etc.
In a system that's structurally rather close to Original Formula Italian fascism, how "private" nominally non-government companies are when so much of what they can and cannot do is highly debatable, to the point where I think your quibble is very very minor.