Tbh, the way I've come to understand it is that Ulfric is useful in the sense that if he wins and drives out the empire then Skyrim will be ripe for the Thalmor to just come in and curbstomp them and take Skyrim by force due to the simple fact that the Stormcloaks are just woefully weaker than the Thalmor forces(...)
(...)Ulfric is just a useful idiot to them because if he wins, they win.
I dunno... I've done both sides in the civil war and I always go Stormcloak every time I replay Skyrim lately. I can't side with the Imperials in good faith. I remember reading about how Morrowind has an aspect of Tiber Septim show up and he explicitly states that the Empire is old and is probably overdue to be reborn. The Mede dynasty taking over doesn't really count.
I think the Last Dragonborn siding with the Stormcloaks just makes more sense in an overall inter-game story way. At the start of the game you're already with Ulfric and about to be executed alongside him. This is already incentivizing a certain camaraderie with the Stormcloaks, and a distaste for the Imperials who were gonna chop off your head. Also, the LDB is a stereotypical Nordic berserker on the cover, and so forth.
I remember that there's legitimate in-game documentation also explaining that the Thalmor see either side winning promptly as a huge problem. They probably do want to conquer Skyrim with that methodology as you said, but the Thalmor are explicitly just too busy rebuilding their strength after the consecutive wars against the entire Empire and then the continual resistance in Hammerfell. If the LDB wins (hastening the war), the Thalmor still lose, or at least have their initial plans ruined.
This is headcanon tier shit, but as an additional I just can't imagine that there's 2 Thu'um users in post-stormcloak victory Skyrim and the Thu'um
isn't gonna be popularized and expanded on for battle purposes. Ulfric is blatantly going against the Greybeards' teachings that the Voice is meant to be used purely for honoring the gods, he used it to kill Torygg and he actively uses it in battle. That definitely sounds like the type of man to teach others the Voice, to bolster his own forces. The LDB could also (probably?) teach others.
Obviously, the Thu'um is a massive force multiplier. This would (hypothetically) make a free Skyrim, which is already a highly defensible country, a pretty strong military power. Historically in the backstory, this was already a truism before using the Thu'um for battle was de-popularized after the defeat at Red Mountain. The Thu'um returning to the common warriors of Skyrim would actually be a return to form in this way.
This would make sense out-of-game too. TES:V is the most popular ES game, and they added gayer versions of the Thu'um powers in Starfield. Of course, the next game is probably going to be in Hammerfell so if any "cool new game-specific super duper power" is going to be included, it'll probably be something like Swordsinging, but the point I'm trying to make is that the Thu'um is one of the most pop-culture/normie memorable things about Skyrim and thus Bethesda has a reason to include it in the next game so people feel comfortable and familiar.
Anyway, siding with the Stormcloaks also lets you kill more elves so that should be reason enough, honestly.
EDIT: typos.