Remember political parties do not exist to the US Constitution! Political parties have no constitutional standing, they are only instruments of political convenience.
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.