The Greybeards are intelligent, but I don't think they have the power of foresight.
How could they know Ulfric would use the Thu'um to start a civil war? Obviously at the time they accepted him, he was deemed suitable. That was a mistake, but everyone makes mistakes. Even wise-old-Thu'um-wielding-beard-wearing monks.
Oh, absolutely. I just get tired of hearing the "Ulfric took a vow and broke it" angle as proof of what a despicable person he is. If he took a vow in order to be
accepted for training, it was an oath he took as a child or teenager at most unless he is far older in 4E 201 than he appears. I put him in his late 40s, perhaps 50ish, but I think 55 would be pushing it. Even accepting an age range of 50-55 in 4E 201 puts him at 20-25 when the Great War broke out and 10-15 years old when he began his training with the Greybeards. If he's younger than 50, we're talking about a child under the age of 10 being asked to make a lifetime commitment to a way of life for which only a small handful of people are well-suited and who then gets lambasted for changing his mind about it ten years later in the face of what looks like the imminent destruction of his country and his people.
Seriously, if some ten year old kid takes a shine to you and tells you with complete sincerity that he/she wants to marry you someday, you don't hold them to that ten years later when they're all grown up. And you don't get all "but I trusted you, OMG you betrayed me" about it because that would be utterly ridiculous.
If the Greybeards can be forgiven for not knowing that Ulfric would grow up to be someone who couldn't stay on the mountain and live a life of peaceful contemplation while the world burned, then he can be forgiven for not knowing it himself. And again IMO he deserves to be cut even more slack than the Greybeards in this respect because of his age at the time.