Sybille also says that when Ulfric brought up independence in the moot, his words were "just short of treason." So a) it's not true that Ulfric did not try to advocate independence using "diplomatic" means, b ) we don't really know that she's right about Torygg's beliefs because no one else confirms he had any such inclinations, and c) "diplomatic means" just means a trip to prison or to the execution block- as Markarth and Helgen demonstrate. If Torygg were at all interested in independence, then he should have been the one to approach Ulfric about it, not the other way around. The buck stops with him, or it should, if he were a real king rather than a puppet. This logic makes no sense to me. If everyone, even Torygg, agrees that Ulfric could win anyway, then I don't know why people assume that Ulfric used the thu'um to "cheat." He obviously had other intentions by using it. I believe it was to show the Nords that they have power in their ancestral tradition that they're not using because reliance on the empire has made them apathetic and weak. "Let the legion do it." Well for Skyrim now, it's a matter of the nation's survival that they learn they can only rely on themselves when it comes down to it. edit- Also probably to help his case that he's the right man for high king.
We don't know the details of that moot. For all we know, Ulfric's words were along the lines of speaking about how much he wanted the Emperor/the Empire to die, which is pretty close to treason. "We should separate from the Empire," doesn't sound like it would be considered "just short of treason" to me, which makes me think that it was probably much more heated (and with a much greater threat of violence) than what you're painting it as.
And what does Markarth have to do with the Empire at all? The deal for Talos worship was made by Jarl Igmund, and never had the consent of the Empire. When the Thalmor found out about they put pressure on the Empire. The deal was never an Imperial deal to begin with. You can't blame the Empire for going back on a deal they never made. That's all on Igmund.
And regarding the use of the Thu'um, I'm not saying it was cheating. I'm saying it was senseless overkill. He didn't need to use it, but he did because he knew it would make the duel more traumatic. It would put fear in the hearts of his opposers, since at a moot, if he didn't approve of what they were saying he could kill them with a single word. He was trying to intimidate and bully his way to the position of High King.
Was a call back to the olden days of skyrim. The greybeards weren't around, a lot of nords used shouts in war and the leader commanded through strength. In fact the nords were a rather dangerous group, known for warriors that would seige their enemies with shouts and were rather successful in most of their efforts(Until they went up against the Tribunal). He broke the greybeards teachings, but the greybeards do not dictate law. Nor was the greybeard philosophy the only one until recently. The empire was supposed to have a college of the voice in Markarth.(Something I really wish they had finished)
They may not dictate law, but what he did was in direct opposition to their teachings, and shows that he's not really grateful for the power they gave him, at least in the eyes of someone like me who watches a lot of kung-fu movies where abuse of a technique by a sociopathic student is a bit of a staple plot.
As a side note, that College of the Voice sounds really cool.