I agree that much more could be done to make spending time in Whiterun worthwhile. But don't think deposing Balgruuf as jarl would do that very well. To me, the Dragonborn should be more of an adventurer than a ruler. Keeping the Dragonborn as a thane would enable him/her to go anywhere and do anything on Balgruuf or the Greybeards' behalf. But being a ruler of a city or province limits mobility if you play the role right. Holding court is pretty boring compared to fighting dragons, bandits, or traitors.
The mod Capital Whiterun Expansion does something like this. Beside expanding the city's neighborhoods, CWE adds more thanes to the city. Balgruuf's thanes work on his behalf to defend Whiterun. Having only just started playing it, I am not sure how far its quests go. But I have completed a battle with the other thanes against a suspicious band of thieves and cutthroats using catapults and battle animals (elephants and sabercats). Maybe they were irregular units allied with the Stormcloaks. This mod author is doing something similar for Windhelm and Riften.
If modifying the city sounds too complicated, you could just focus on building up the Whiterun thanes or all thanes as a faction. When I first started playing Skyrim I used to have a mod that added a thanes' hall outside the city walls, I could go there to get followers to help in my quests. Can't remember its name. It was not available on the Nexus (just through the Steam Workshop). Wish I had that mod now for Special Edition.
The ultimate prize could be some kind of uber thane title over all of Skyrim bestowed by the high king or whichever side the player favors in the civil war. Or maybe the Greybeards make the Dragonborn some kind of champion of Skyrim since the land is without its high king at the moment.
What do you think? Am I off my rocker or does this make some sense?