Ulfric did want to do it the peaceful way. He gave away his chance of a surprise attack, or at least an attack without the imperial forces present, by giving Balfruuf a choice. You can hardly blame him for "going after" Balgruuf when the jarl calls in the imperials. From the dialogue with Galmar you can learn he does so only with great reluctance. He even spares Balgruuf's life after the battle.
All Jarls have their lives spared by everyone. That's just a game mechanic and doesn't mean much, honestly. And don't read too much into what Ulfric says his reasoning is. He's a big showman and wants to come across as doing everything for a free Skyrim and for Talos worship, but it is pretty dang clear he's not afraid to kill tons of innocents in order to become High King. After all, if he massacres a bunch of innocents, it's the Empire's fault and that is that. He likes giving speeches though, as other people comment on.
He forces the issue with Balgruuf when he knows Balgruuf doesn't want to get involved. And it is quite clear that Ulfric has been maneuver troops to attack Whiterun for some time. The fact is Ulfric is for a Skyrim not free to rule itself, but a Skyrim that he rules. There's a big difference between the two.
In times of great turmoil, neutrality is a choice. And indecision can be a major flaw. Countries like Belgium and the Nerherlands thought they could be "neutral" in WW2. They were wrong.
This isn't anything like WW2.
Balgruuf, like any man, has the right to stay out of any conflict that doesn't concern him. The more so if neither side has a monopoly on virtue. But the Civil War does concern him. His people are arrested by Thalmor patrols. His country does belong to Skyrim and the results of the civil war will also be the results for his people. In such cases indecision and neutraility are almost a crime, at least a mistake. As I posted on another thread: "For evil to triumph, all it takes is that good men do nothing." (Burke)
The Civil War only concerns him because Ulfric wants to rule instead of the Empire and is pretty willing to do whatever it takes to make all of Skyrim bow before him. That's honestly the behavior of a tyrant. Not wanting to bend his knee to such a person is quite understandable.
Heck, the only reason there ARE Thalmor patrols in Skyrim is because of Ulfric's past actions as many people state. Before the Stormcloak Rebellion started, there were no crackdowns on Talos worship.
Because the empire was so helpful when Morrowind was being invaded by the Argonians?
It's funny that people seem to have more trouble with the idea of Ulfric attacking Whiterun in order to win his war than the imperials allowing Thalmor to kidnap its citizens for being Talos worshippers. If you agree with the ends, the means are just what you have to do. Ulfric does not attack Whiterun lightly or with bloodthirst, in fact he's quite reluctant.
He's not that reluctant. He's positioning troops to do it and making speeches about how it's a shame and he'll do whatever he feels he must (to rule). He then forces the issue with Balgruuf. This really just makes him a hypocrit, since he's just as bad as he claims the Empire is -- and he's quite willing to overlook the massacre of civilians by his people.
And honestly, there's no particular evidence he's going to do much of anything against the Thalmor beyond not have them in Skyrim...which was pretty much the case before he started his rebellion anyhow. At best things will end up much as they were at the cost of a lot of lives. At worst he'll keep Skyrim in a perpetual war with the Thalmor just like Hammerfell is, with countless lives lost.
I just don't see what his worthy end is supposed to be.