The problem is the Imperials didn't give you the opportunity to prove your innocence.
The Captain herself essentially told Hadvar to screw the list, and that you go to the block anyways. The Imperial supporters don't understand that this is a HUGE problem for us Stormcloak supporters, and this one of the major complaints we have against the Imperial legion in Skyrim. To go around slaughtering people without letting them prove themselves innocent is barbaric. It goes beyond humanity. It's flat out criminal. The Imperial legion went from being protagonist to being antagonist.
Any intellectual person would realize that perhaps Lokir isn't truly a Stormcloak when hes acting like a coward, and pleading for his dear life. You would think that the Imperials would realize that he isn't part of the Stormcloaks, and that he should only be fined/ and imprisoned for stealing the horse, but no they wand up murdering him without probable cause. If the Stormcloaks didn't back him up I'm certain that his family back in Rorikstead would've vouch for his innocence.
You have to also factor in that the Imperials know the Thalmor are behind what is happening in Skyrim, the General knows, officers know. Your word doesn't mean that much when it comes to the larger picture. The head of the Thalmor in Skyrim showed up at Helgan. When they're snooping, you must be certain you end their plans.
This is one of the issues for Stormcloak supporters, small minded not looking the the larger picture, the Thalmor are behind the Civil war in Skyrim, they started stirring up trouble, they're aiding the Stormcloaks. Put yourself in the Legion's shoes, you have a bunch of prisoners that the Thalmor suddenly take a deep interest in, you know the Thalmor are behind the rebellion, you know the goal of the Thalmor is to destroy the Empire, now they're very interested in your prisoners, they don't want there to be an execution because it would harm their plans. The prisoners were originally being taken to Cyrodiil for trial but a slow moving prison transport several weeks from Cyrodiil? You really believe the Thalmor would simply let their plans come to an end because a very slow moving transport is very slowly getting away. Tullius is no fool, it is very clear the Thalmor were keeping a close eye on the transport because they made it to Helgan very quickly when the Legion suddenly diverted, they were waiting for Tullius, they exchanged some words though we couldn't hear them, no doubt it would of been about stopping the execution because they need Ulfric alive and distracting the Empire, making sure the Empire wastes resources and weakens their army on the Aldmeri Dominion border by sending in Legionnaires who are stationed on the border. The Legion in Skyrim has had to recruit locally because the Empire can't weaken their defenses too much.
"Most of the Legion is tied down on the border with the Aldmeri Dominion. The Emperor can't afford to risk weakening Cyrodiil's defenses. From the Imperial City, our war here is just a sideshow. An interlude before the main event against the Thalmor resumes." - Tullius
Ulfric: "Stop what? Taking Skyrim back from those who'd leave her to rot?"
Rikke: "You're wrong. Ulfric. We need the Empire. Without it Skyrim will assuredly fall to the Dominion."
Tullius:
"You realize this is exactly what they wanted."
Galmar:
"What who wanted?"
Tullius:
"The Thalmor. They stirred up trouble here. Forced us to divert needed resources and throw away good soldiers quelling this rebellion."
The Aldmeri Dominion is something that Skyrim cannot beat alone, they can't even face all out war with the Empire, yet they plan to attack the Aldmeri Dominion?
"We're ready to march on Solitude, but the Emperor's cousin is getting married! If royal blood was spilt, all of Cyrodiil would be up in arms. We can't afford an all out war with the Empire. So we'll bide our time for now." - Ulfric Stormcloak
"First we'll kick the Thalmor and their bloody Imperial puppets out of the country. Then we'll rebuild Skyrim into the land she once was. When we are done with that, we will take our army to the Dominion, and show those pointy-eared bastards not every man is fit to be their slave." - Galmar Stone-Fist
"Because getting rid of the Empire was only half the problem. Soon, the elves will again seek to rule the world. We must ready ourselves to fight them. For it will be Skyrim that shall lead Tamriel in those dark days, when the fate of the world is finally determined." - Ulfric Stormcloak
The real threat is the Aldmeri Dominion and the only way to defeat that is if man is united, that will include Hammerfell which hopefully will be able to mend their relations with the Empire once more.
There can be no doubt that the current peace cannot last forever. The Thalmor take the long view, as is proved by the sequence of events leading up to the Great War. All those who value freedom over tyranny can only hope that before it is too late, Hammerfell and the Empire will be reconciled and stand united against the Thalmor threat. Otherwise, any hope to stem the tide of Thalmor rule over all of Tamriel is dimmed. - The Great War
You claim you like Tiber Septim, yet Stormcloaks are destroying his Empire, disbanding the Third Empire and thus ending the great Empire of Tiber Septim. Skyrim is not only strategically important to the Empire currently, it is also symbolic, Skyrim and Cyrodiil conquered Tamriel together, the removing of Skyrim is the real end of Tiber Septim's Empire.
"Because even if we can't openly worship him, Talos the god was once Tiber Septim the man, and this is his Empire. And Torygg wasn't ready to let it fall apart." - Sybille Stentor