...so I was thinking about a neat little trick that might be able to build a more cohesive country.
Imagine if the outgoing President remained in the Cabinet regardless of party affiliation or relationship with the incoming President. This way, leadership would have the chance to pass down traditions from one term to the next, and prior Presidents would still have a token, artistic, organic say as to how the country is run.
The DoI isn't terribly powerful, but it's still something that needs to be taken care of. It would also ensure that Presidents literally don't forget about the land they expected to govern over. If things don't go well while Presidents, they'll be held responsible while Secretaries.
Another advantage is the DoI could serve as a cooldown responsibility for Presidents re-entering public life. They could go on a relatively neutral campaign, supporting and touring landmarks, and just being generally patriotic. In the case of questionable policies, it would also give ordinary people a chance to ask Presidents to atone and confess for whatever they did.