What bonuses do you get? I don't see where they are displayed.
We in fact have events to urge you to move to capital, I do not know if there's event to move back though.
If mobs of Constantinople were more aggressive...
Edit: In a Byzantine-style empire, I think that burghers of capital, especially Constantinople, should be more "privileged", as they may expel you before you could raise an army to suppress them. Not only did they riot in Byzantine empire, but they also riot in Ottoman empire.
The capital itself should also be more important. Not only the 4th Crusade, if capital of a admin empire is sacked not in a civil war or claim war, the empire should just be shattered. The capital is shaped to be the center of its bureaucracy, without it as a symbol, local ambitious generals may want to carve their own realms.
So lucky Honorius's capital was in Ravenna and could not be used as an example.
Besides, influence is in fact a quite abstract resource. Does that mean your popularity among bureacrats? Or does it mean you populariy in Constantinople? A noble could easily be loved by fellow bureacrats, but hatred by local burghers, and a unpopular Basileus could easily be expelled from the city, no matter how influencial within the empire.