I think it would change the political landscape in ways we can’t even imagine.
Seeings as how the VPOTUS is almost worthless - it would make the VPOTUS even more worthless.
Currently the VPOTUS serves as a back up man to the POTUS as far as fund raising and ideology - speech making - the ‘bully pulpit’ and whatnot.
If the VPOTUS was separated from the POTUS it would ONLY have the Constitutionally mandated role - which absent the inability of the POTUS to serve - is almost NOTHING - as a tie breaking vote in the Senate is an almost negligible power, and being “President” of the Senate is purely a formality.
Anyone trusted with the power of the POTUS should, Constitutionally - also be trusted with selecting an ideologically aligned and eligible successor.