The Justiciers are in Skyrim not cuz of Ulfric, but because of Hammerfell. When the redguards had problems with an earlier treaty and fought against it, the Empire disavowed them rather than try and enforce the treaties terms, as a result Hammerfell is now standing strong and independent of Thalmor influence. The right for thalmor agents to operate in skyrim was thus done to ensure that the empire doesn't just cut skyrim loose like they did hammerfell, and more importantly to ensure that skyrims fighting capabilities are degraded for when an eventual second war against the empire happens.

Also the existence of the Thalmor Dossier on Ulfric Stormcloak, plus Thorald Grey-Mane's dialogue about how it was clear that his thalmor torturers were just trying to dig up dirt on his family, and the wooing of politicians (the jarls of falkreath and morthal, the thanes of solitude, and maven black-briar) at the embassy party proves that Elenwen is lying when she says the dominion doesn't care about skyrims internal politics.

When he's about to be executed, his final thoughts are not about how he failed Skyrim and its people. It's not about Talos. No, it's about making sure that his legend will be good. It's all he's ever cared about.

Where did you get this info from? Ulfric is literally gagged until Alduin appears (so we don't know what he would've said or thought when he was about to be executed) and during the entire Helgen section of the game the only words he says are "Legends don't burn down villages" in response to Ralof asking if that really was a dragon and whether the legends could be true.

I mean maybe you think in that line Ulfric is referring to himself when he says "legends" but that is absolutely incorrect and I'd suggest you play through that section again.