I can't picture him barely talking and wearing robes, being peaceful and showing self control.
His fate changed so much from when he was just a boy studying at the throat of the world.

I think of him as someone who probably tried, and might've wanted that at one time, but was also torn between his own upbringing in politics, as a noble, etc.. Long lonely nights on High Hrothgar where his mind kept slipping into ideas about how what he could actually do with his power, and thinking that the Greybeards might be a bit misguided in their passivity and uninvolvement of human affairs. Nothing wrong with that either. I'd think the same. I just think it got to his head too much. There's a difference between helping and getting involved with people, and thinking that you're the true high king Skyrim needs. I don't care who you are. Those are never good thoughts. If he was to become king, it'd better if it just happened, and that he didn't try to force the issue.

Maybe he was kicked out because the Greybeards noticed an unhealthy ambition is his mind.

The Jarl of Whiterun also made a pilgrimage to the monastery and saw the Greybeards though he doesn't mention being taught anything. He seems more respectable and less self-centered than Ulfric.

I guess I kinda figured he did what he was told to do and liked that it was a great calling until he realized he really didn't like it... and then he left and went elsewhere. Not to say he couldn't have done it, just that I think it would never have been what he was really good at. He's a leader of men and he's a warrior. They scripted and annimated him really well in that.

Maybe he was thinking if Tiber Septim can do it so can I.

Ulfric was sent to be trained as a Greybeard at a young age, he had no choice in the matter. He does not say as to why he was sent, just that he was.

He was going to continue to become a Greybeard, but hated the thought of sitting back and doing nothing for his people, so he left.

I doubt the Greybeards would kick him out, does not seem to be something they would do.

Ulfric as a Greybeard, I cannot see it happening, but a lot of people do not assume a person cannot be this or that by what they are now and not understand people change. They were not always who they are, normally its an event that changes the course of a person's personaility and their destination.

That's another thing. I think he's represents Wulfarth's personality more. More militant, more ambitious, more divisive with race than Tiber Septim was. And in one story, Wulfarth heard the call of the Greybeards for the dragonborn, and he was told he wasn't it. I wonder if Ulfric entertained the idea at any point.