What would be the point of electing the Prez and the VP separately? It changes nothing to be honest. You kind of want the VP to be from the same party as the P.
The one thing it would lead to is shenanigans regarding a president who has committed malfeasance or is undergoing a disability.
Assume it came to light Trump really had colluded with the Russians to engage in fraud in the 2016 elections and Tim Kaine had been elected Veep because Americans like to ticket split. The Kavanaugh Supreme Court appointment is on the line. Would Republicans expeditiously without dragging their feet and uniformly be more or less likely to remove Trump from office?
Assume in 2020 Joe Biden had been elected President but Mike Pence is the Vice President. Wouldn't at the current time Mike Pence be agitating to remove Biden for disability? Would the Ds be more or less likely to remove him if Pence would become the President? Seems like a recipe for "Weekend at Camp David" to me.
If the Veeps are from the same party as the President, the results will act as a separate power base (a shadow cabinet) from which the President can be checked. If the Veeps are from a different party, it will lead to shenanigans where the opposition party will pretextually try to remove the President. In the Soviet Union and in some Latin American countries, it's been enough to form a power base from which a coup can be launched.