That logic is ok until we consider that Ulfric may actually BE a Thalmor agent as his dossier says (broken by Elenwen and assigned to her as an asset only to grow uncooperative of late) in that case he probably wouldn't support the empire against the common foe at all if the destruction of the empire is the ultimate aim.
Ultimately everything we are told about the civil war's origins starts with Ulfric as the source.
Hod mentions "everyone had their little shrine to talos" and that enforcement of the concordat was practically non existent until the Stormcloaks started agitating about it.
Then we also had the Markath incident where Ulfric captures Markath and doesn't hand it back unless he gets his demand of free Talso worship, the jarl of course wanting his city back foolishly agrees. The Thalmor find out about this as was planned and send in the justiciar inquisitors to create even more resentment and support for Ulfric's cause so that he can launch his civil war.
in such a situation of a stormcloak victory and skyrim leaving the Empire Cyrodiil would be greatly weakened and ripe for conquest as much of the legion is made up of Nords, Ulfric might also be worried that once he wins the war that the Thalmor might double cross him and attack Skyrim anyway after they are done with defeating the remnants of the Empire.
The part about mentioning ""a stormcloak victory should be avoided" probably hints to the Thalmor thinking they cannot control Ulfric as a puppet and he will probably turn on them once he is high king. (and that they also want the sides in the civil war to be killing each other for long enough to weaken Skyrim as well as Cyrodiil for conquest)
Makes for an interesting thought.


