The Empire's somewhat caught between a rock and a hard place, in truth.
Even if Tullius is the only "problem solver", he was given command of a Legion that was ill equipped and not even close to being at full strength (hence the need to recruit locally). His skills as a general allow the Empire to stalemate the civil war in terms of open conflict, whilst very nearly managing to end it anyway (Before the intervention of multiple 'divine' beings).
So we keep this in mind.
The Empire has at least six to ten Legions outside of Skyrim; that's obviously more than enough force to crush the rebellion. Depending on how you handle the civil war and Dark Brotherhood, Ulfric will admit as much. The Stormcloak cause benefits from it being treated as a sideshow.
So why not just send more Legions and crush the rebellion, instead of sending a problem solver like Tullius with limited resources? Great War 2.0 is just around the corner. Both sides acknowledge preparations for it. If more legions go north, it's practically giving the Aldmeri Dominion an open invitation to initiate hostilities again. The Empire knows the Dominion has eyes (and they would anyway, regardless of the White Gold Concordat before anyone tries to say otherwise) watching so this troop movement won't go unnoticed by the Thalmor.
A rock and a hard place indeed.
He was only really right about the Jagged Crown. I don't really recall any other moments of significance, though it has been a while.