You're right.
I was thinking for some reason that it was the opposing party candidate that got to be Vice President.
But it was actually the President and Vice President running independently. So you could still have the Vice President as the same party.
In other words...it wasn't the loser of the Presidential race who became Vice President. It was the winner of the Vice Presidential race who became Vice President.
Anyway...I still think it would be a cool idea.
And it would take away the executive branch having that tie-breaking vote in the Senate if the Vice President was from the opposing party.