And the flip side is that if your central authority is weak and corrupt, they drag the whole shebang down with them.
Beyond a strained military, there's really no evidence for that. Well, perhaps the Emperor being weak in his old age. Where's your evidence of corruption though?
The problem here is that ethnic imperials have become the ruling elite, and they look only to their own interests and to protecting Cyrodiil. What's past is past.
They don't seem any more the ruling elite than they were in the past. Where's the evidence that they only look after their own interests and protecting Cyrodil? The whole Empire would have suffered if the Emperor had been killed, so if that's what you are talking about, it seems like a pretty weak argument.
Jarl Igmund contradicts that. He says that they promised Ulfric and his men freedom of worship in exchange for their help in retaking the city. He then decided to imprison Ulfric and his men, since the Thalmor found out about the deal.
That's not at all a contradiction. Someone can ask you to promise them something. Nor is the Jarl clear on the timeline. Basically he doesn't say who came up with the idea or when.
All of that would only be band-aid measures against the real problem. The Thalmor are in Skyrim because of the empire's capitulation. The empire is acting as a shackle that allows the Thalmor to attack Nords who defy the Talos ban. Loose the shackle and you're free to fight.
It's stated by more than one person that the Thalmor are in Skyrim because of the rebellion, because Ulfric made a huge deal out of Talos worship so now the Thalmor have the excuse they need to move in and crack down. Ulfric has done exactly what the Thalmor want him to do -- this is uncontestable and in the Ulfric dossier. The man is helping secure the shackles. Even if he wins Skyrim, he's greatly weakened Skyrim's military, divided the people, and weakened the Empire.
And a band-aid measure? Slowly bleeding your enemy dry via guerrilla warfare can be extremely effective. It's worked throughout history to get rid of unwanted influences. Further, you don't have to kill your own armies or people. Ulfric can't say he couldn't think of the idea, as Skyrim has people in it that do this (the forsworn). But Ulfric clearly wants to be High King as many, many people comment on in the game. So he picks a strategy for whether it nets him High King of Skyrim over whether it avoids killing a bunch of potential allies and a bunch of his own people he CLAIMS to care so much about.