"Technically, a new President isn't needed if there's a veto-proof majority in Congress"
True, but that supermajority is 2/3 majority in BOTH houses. The last few years every single topic seems to have become extremely polarised no matter how trivial or 'obvious' the issue is. In practice the only way to have a veto-proof majority is for one party to control 2/3 of both houses. And, also in practice, if one party has a 2/3 majority in both houses, they will also control the presidency so no veto-proofing would be a moot point.
The current situation with split government, the only way to get anything through both houses and teh president to sign is attaching barrel-loads of pork to every bill.