What if the 2nd place lost by a landslide of mega proportions? It would be better then for the VP to run on their own in the same fashion as the President.
Also I would have the states vote for the Presidency individually. The voters in each state would vote for the president, then the result would go to Washington as a vote for that candidate. A tie would be determined by a total sum of all the votes in all of the states. This would give every state equal representation.
But I agree with your fantasy of no "parties" because factional representation always is unjust.