His son who is vastly superior than his father in every way. He has better stats, is a better general, and has no detrimental vices comes to power and he still gets the same lousy influence his father had to start.
Well, YOU know that, but everyone else only knows what they've been told. A new head of state on the scene is an unknown, preceded by rumours and gossip at court and maybe some tales from vanquished foes. Who knows if he'll continue his father's policies? Who can tell how many of his father's men will follow the son? Which alliances will he maintain? Which allies will remain loyal, which will make a land grab?

Managing a succession can keep your nation's eyes off the outside world. There are loyalties to confirm, barons to reassure, landowners to appease.

It's a pain, but I think it does reflect the uncertainty of such transitions. As for the Byz, well the new guy may well be inbred, drunken and cowardly, but he IS still the Byzantine Emperor, drawing much of his "influence", not from the current state of his empire but its glorious past.

Of course, once the new chap settles in and he gets his ass properly on the throne, his influence will shoot back up.