User blog comment:Harold Burned-Mane/The Truth about Ulfric Stormcloak/@comment-173.49.68.39-20140720214709

I just finished watching the Stormcloaks questline on YouTube and you are right about Ulfric not wanting to be High King because he says in his speech after the Battle of Solitude "But it is all of you who are the true heroes! It was you who fought a dying Empire who sunk its claws into our land, trying to drag us down with it. It was you who fought the Thalmor and their puppets who would have us deny our gods and our heritage. It was you who fought your kin who didn't understand our cause, who weren't willing to pay the price of our freedom. But more than that, it was you who fought for Skyrim, for our right to fight our own battles... to return to our glory and traditions, to determine our own future! And it is for these reasons that I cannot accept the mantle of "High King." Not until the Moot declares that title should adorn my shoulders will I accept it."

I think this is reason enough on why he doesn't want to be High King or he says this to contradict himself so he can become High King, but we may never know until Elder Scrolls VI.