Even if it happened, your loyal constituents would still be sending you out to do their laundry and fetch troll dung from the near by caves. Honestly, I don't even see what the point of being the leader of everything is. Since you don't actually lead anything.
Maybe they just hate their new Jarl?
But seriously, if the player could become Jarl, the best option would probably be the one town that doesn't haven one anymore: Helgen. (Because not everyone is going to play for the same side in the civil war and booting someone else probably just isn't feasible.) The only problem: It's a mess. Be in charge of the mess. Get builders to make a mill, have the mill make wood for rebuilding, mine ore for rebuilding, recruit people to do all the boring parts like farming, smithing, chopping wood, etc. Basically the Dragonborn would be recruiting potential villages and overseeing work. He or she could participate if you
wanted to.
I've thought about this too much...