You are partially correct , at one time the office of the President and vice- President were separate offices and ran separate campaigns. The “President did not choose the vice- president. If this was still true , we could of had biden running a separate campaign against Palin. We could of had obama for the president and Palin for the vice-President. The problem is obama did choose biden and if his campaign is voided, all actions he took during the campaign is also voided( his selection of biden). biden knew obama was not eligible to be President , supporting his election was a act of treason.
Constitutionally (therefore legally!) there is an office of the president and a office of the vice president.
How we have morphed this into a single entity is constitutionally irrelevant. Now of you want to argue that Biden was a willing accomplice then ok, but that's a different discussion.