I do not believe that is true. When that was written we voted for VP separately. We voted for a ticket, so if the top of the ticket goes so goes the ticket. At least I would argue that.
We vote for the ticket, but the electoral college does not. Congress also handles them separately.
Your argument doesn't hold water. As far as the Constitution is concerned, only the vote of the electors matters, and the electors continue to vote seperately for VP and President.