That logic would make the Republican Plan and Obamacare unconstitutional. Medicare would also be unconstitutional. Obamacare has passed the constitutionality test in the US Supreme Court. A single payer plan like "Medicare for All" would not be unconstitutional. A side topic: Medicare works well and includes options for private insurance like gap coverage. And a large percentage of  people have their medicare coverage administered by a private insurance company (an "Advantage" plan). I think Medicare makes a good vehicle to build single payer around.