This. He didn't trick the Greybeards. He just betrayed their trust after the fact.
Then their trust was based on the expectation that the child they chose to train would grow up to be who they required him to be, not who he actually turned out to be. If they really believed that was something that they could control, or that Ulfric was obligated to them as an advlt because of decisions that were made for him when he was a child, then they are not as wise as they appear to be. If they were certain that Ulfric would grow up and still want to become one of them, and that's why they chose him, then they made the wrong choice. That was their failure, not Ulfric's, and he is not obligated to them simply because they screwed up (if indeed they even feel that way about it).
As I've said in other threads, unless the Greybeards are willing to keep their students under their immediate control (i.e., physically confined to High Hrothgar) for their entire lives, by force if necessary, they cannot ever be sure that what they teach will not be used in ways of which they don't approve. They have even less expectation of that if they take a child to be trained, because the day will come when the child will be an advlt who is free to walk away from the life that someone else chose for him. If they assume otherwise, then they are trusting in the inevitability of an outcome that they themselves cannot guarantee and which the student cannot guarantee as a child and is not obligated to give them as an advlt. That is foolishness, and the Greybeards are not fools.
I think it's fair to say he disappointed them, but "betrayed" implies an obligation that never really existed IMO. An expectation is not an obligation, and children are not bound by the expectations of others once they become advlts and are free to choose for themselves.