Epiphany Trek Edit
The Federation President is a siting member of the Federation Council elected by their fellows to chair the Federation Council. They are not elected directly by citizens of the Federation. Indeed the average Federation citizen might be hard pressed to tell you who the Federation President is. That person has far less effect on their lives than their own City Manager, which they pay closer attention to, or the leadership of their planet.
While Federation President is a position of some responsibility it is closer to the President of the interstellar rotary club than POTUS as often depicted. Federation President does not have the power of a head of state, they are not one. Under the usual situation the delegation of that planet is run by the under councilor while the councilor serves as President.