The Stormlockas don't want The Empire destroyed, they just want to be free from it, even they know the empire is needed to fight the AD, and with The Emperors death, it means more ploitical unrest, and the AD having a better shot at killing everyone. The Emperor being killed is not good for them.
Furthermore, if you try to invade Solitude, and take it over for the Stormcloaks as part of the civil war, when the Emperor is visiting, Ulfric tells you he wont order the attack because he knows if royal blood is spilled during a Stormcloak attack, the Empire could actually get serious and face-roll the Stormcloaks.
So not only is the Emperor's death bad on the AD front, but it also gives the Stormcloaks reason to fear The Empire actually getting serious about the civil war, and face-rolling them.
And the emperor is technically the emperor of Skyrim, the emperor of Cyrodiil, and the emperor of High Rock, he is, by being emperor of The Empire, emperor of all provinces within the empire.