I think influence only impacts who is automatically selected as faction heir by the AI. It also had an impact on Senate offices in vanilla, but those are gone now. So, I suggest (and I'm almost certain that the EB team has probably already done this for their internal builds), that there be two more traits:

First Born Son, and Son of the Pharoah/ Basileus, King, whatever it is for the faction.

The first born son trait is influence +3 and the Son of (Leader) is influence +2. That way, by merely being born, they have a greater chance of being automatically selected by the AI.