PS. I don't think the Greybeards approve what Ulfric did. They trusted him, that's why they tought him in the first place. He betrayed that trust.
Ulfric himself admits that Arngeir has probably never forgiven him for what the Greybeards would no doubt consider his "blasphemy" in using the Thu'um for anything other than the worship of Kynareth. He also states his reasons for leaving High Hrothgar when he did and for having fallen from their strict discipline. I found the whole conversation rather moving, especially given his overheard remark that he would "gladly retire from the world" if the day came when warriors were no longer needed. He also talks, somewhat wistfully IMO, about how HH is so "disconnected" from the world of action and conflict.
My interpretation has always been (even on first playing through that conversation) that he left High Hrothgar with some regret and still feels badly about it, but that he believes the Thalmor and Dominion were and are too great a threat to be ignored. He had to choose between the path of action and the path of meditation, not as the child who first climbed the 7000 steps but as a man watching the world he loves torn apart beneath them.
And please note the "as a child" part of that last sentence. Ulfric speaks of being chosen by the Greybeards when he was "just a lad," i.e., still a child. Whether this means that they sought him out on their own accord, or that his family put him forward as a candidate for training and the Greybeards chose to accept him, or that he himself wanted to join their order and both his family and the Greybeards agreed to it... the exact mechanism of how he ended up at High Hrothgar as a child is unclear. In any case, he was still just a child.
So maybe the Greybeards need to have a little accountability on the issue of Ulfric's "failure" to adhere to their teachings over the long haul. He was a child when he was called, or sent, to High Hrothgar. They were far older and presumably wiser. They are the ones who chose him, or at least accepted him for training, at an age when it was impossible for
him to know whether or not he would grow into the kind of person for whom that life was appropriate or acceptable or even tolerable. It would've been impossible for them to know as well, but they were the advlts in the equation. IMO the bulk of the responsibility for accepting an ultimately "unworthy" student falls on them, not on the child who had the nerve to grow up and fail their lofty expectations. If the Greybeards never considered the possibility that the child they chose might turn out to be a man of action and not the passive, meditative navel-gazer they were hoping for, then that's
their failing, not Ulfric's.