UnitedHealth CEO shot

Correct, but its closer to what single payer looks like than purely private health insurance (hence the Medicare for All movement). Medicare is not trying to ape the private healthcare model - in the case of the Anthem/anesthesiologist kerfluffle it's the other way around.
it seems hard to draw links though - like I don't think we should assume that public education that we have in this country would look/work the same if there weren't private school options - I think public education would be very different if it was universal