Gotta say, Yahtzee, I find it a bit odd that I didn't even play the first Darksiders game and yet I seem to have a better handle on the story than you do.
Indeed, in the Darksider universe, the Horsemen are not the living embodiments of their namesakes. Turns out they're just 4 warriors of a warrior-race that was creating havok across the cosmos. These 4 saw the destruction they had wrought with their brethren and decided that someone had to put a stop to it. So in exchange for ridiculous amounts of power, they sign up with the Cosmic Bureaucracy to serve The Balance and try to maintain said balance across the various planes of existence. As such, they essentially become the strong-arm enforcers for the Bureaucracy, going around punishing those that break the rules of the balance, including a cease-fire with regards to the human realm. Evidently a pact was formed between Heaven and Hell as arbitrated by the Bureaucracy stating that only once the 7th seal has been broken could the two oposing armies meet on Earth and duke it out. However War was tricked into starting things early which lands him in the comsic brig.
So now we come to Death. His story apparently takes place during War's imprisonment at the beginning of the first game. As for who Death is and what are his powers, yes, he's just a guy that's really good at killing things - just like his mates - however he does have the powers of the Reaper. What you apparently missed, however, was that just like War before him: Death is not riding with the authority of the Bureaucracy behind him. As such, his power and authority is severely diminished. This explains why you can't fly around in Reaper Form the entire game: his powers are still sealed away (presumably in the 7th Seal, to be specific). The King of the Dead even tells you this flat out: "You do not ride with The Council's authority. Until you do, you are beneath me." This implies that Death should hold rank over the vast majority of the characters in the game, but since he's acting on his own, he is denied that authority. This leaves him as just being a guy who's very good at killing...who can also tap his deeper power by transforming into the Reaper for short bits at a time.
As for motivation, he believes (and rightfully so, from what I've been told by friends that have played the first game) that his brother is indeed innocent. That War was "The eldest and most noble of the Horsemen, and that he would never have done what he did on purpose." So yeah, you can chalk it up to blind loyalty to his brother. But as we also learn, the 4 are the last survivors of the race that they came from, can you really begrudge them a bit of loyalty towards one another? One of my favorite parts of the game was how through Death's dialogue they actually seem to be mocking traditional adventure/RPG quests. "You must go and open the lava gates so we can light our forge!" is met with Death responding by essentially saying "I'm Death, why the fuck should I do you a favor?" Death doesn't care about the concerns of the quest givers in the game, he just wants to get his own personal quest done and over with. He feels that he should have the authority to demand that the King of the Dead help him with no questions asked, only to once again be reminded that he's riding without the true authority that backs up his power.