From what I can see in the game data, "The Whispering Door" originally had a very different plot.

Considering that Balgruuf's children do not leave with him if he's exiled during the civil war, this quest seems to have at one point explained why.

First of all, originally, the Ebony Blade merely required a large amount of infamy to restore it; obviously, "fame" and "infamy" do not exist in the final game.

Secondly, there are references to a woman named Karinda -- she has a horse in the game files, but she does not exist.

What she would have done is conflicting -- either she flees the city, or Balgruuf's son Frothar attacks and kills her (in a Daedra-induced rage, I assume), resulting in the children's expulsion from the city.

And at some point, the quest required the ritualized murder of Jarl Balgruuf using the Ebony Blade.

Now, this last bit is interesting -- in the game files, Balgruuf is the only Jarl whose base record is not essential. He is made essential by two things; first, his alias in "Before the Storm", and his alias in this quest. "Before the Storm" ends and removes that, but this quest always remains active and never clears aliases.

Due to some leftover scripting, if the Ebony Blade is restored (that is, assuming that stage 100 of the quest can be reached), Balgruuf can be killed with the Ebony Blade, and only the Ebony Blade.

Some screwy scripting then results in Vignar Gray-Mane becoming Jarl prematurely (he acts as though the Stormcloaks have taken Whiterun), but the intended result is that Balgruuf's brother Hrongar would become Jarl. He would give all of Balgruuf's lines for the main quest (post-"Before the Storm"), attend the peace conference at High Hrothgar, and can be given the axe during the civil war questline. I'm not sure if he'd actually take Balgruuf's place during the attack on Whiterun, or if he'd go into exile, though. Tullius and Rikke and Ulfric and Galmar all have recorded lines that reference Hrongar instead of Balgruuf (only the Imperials have a varied reaction -- it's mostly just swapping names; "You think I need to send Hrongar a stronger message...").

On that note, one map scene for the Imperials indicate that, if Elisif were to die (a cut quest indicates that Boethiah's quest would involve killing her), the Imperials would be backing Erikur for High King.

Just some interesting leftovers, and the potential possibility for characters willing to mass-murder their friends to kill Jarl Balgruuf, if inclined.