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Let's use Commander Shepard as an example. Look at Paragon Shepard. He saves the universe, spares as many people as possible, always thinks before he acts, and emerges the better man for it. You know what that is?
When Shepard (or Aragorn, or any similarly paragon-ish character) kills or hurts someone, it's probably because he has no other choice or because it's the objectively right thing to do. And that's boring. No-one cares when Aragorn kills a troupe of orcs, for example, because they're evil and killing them is the right thing to do. Aragorn certainly won't lose sleep over it.
And I don't want to read about that. I want to read about characters who aren't some lofty ideal, who are as human as I am and are thus susceptible to mistakes. I want to play as a Shepard who lets people who didn't need to die die because either a) he was being short-sighted, temperamental, ignorant or any other trait associated with Renegade Shepard or b) this is the real world (from his point of view) and sometimes soldiers like him have to do the ugly thing (Note: This does not have to be true in all cases)
Does that mean I want Shepard to sleep soundly at night, knowing that all those lives are on his head? No. I want the opposite. I want a Shepard who is constantly haunted by the effects of his mistakes and who feels the consequences of his actions even in the present. Shepard does not have to be a good man, but he certainly is a great man. And great men are prone to great mistakes which affect a whole bunch of innocent people. I think it's important, however, that Shepard understands that more than anyone.
Hell, let's use "the hero kills a bunch of guys" again. It could be Aragorn Bestest Hero Ever killing a bunch of utterly unsympathetic monster-men...or it could be the hero killing a bunch of sympathetic characters over a dispute that could have been resolved without violence easily, had it not been for one of the hero's negative character traits.
And that's just one possibility. The antagonists don't have to be evil orcs led by a demonic overlord. That's boring. Maybe the antagonists could even be people just as (if not more) sympathetic as the hero who happen to be on the opposing side. I'll take the Imperials and the Stormcloaks as an example. It's easy to paint people on the other side as imperialistic tyrants wishing to erase a vital part of Nordic culture or as racist barbarians deluded by a power-hungry Jarl. But the more interesting approach is to view them as they are: as people who are willing to stand up for what they believe is right (just like you) who just happen to be on the opposing side.
tl;dr I dislike goody-two shoes characters, prefer instead morally grey characters