Now, that said... games like the World of Darkness series, Firefly RPG... in fact, most skills-based RPGs are designed to accommodate this type of role-playing.
While systems like FATE do have mechanics for the player manipulating the Narrative, that's really in a different sense than is posited in the OP. I don't think just being "skills based" really makes a difference. The issue at hand is simply asking the question - *why* are you killing things and taking their stuff, and what does that really mean, to a sentient being? Exploring moral, ethical, and personal themes is a lot more about adventure and campaign content than about mechanics.
D&D *COULD* do it... but 30 years of "kill things and take their stuff" has burned itself into the heads of every D&D player.
Here's a really big hint - there are tens to hundreds of thousands of D&D players. Attempting to speak for all of them is just a tad much.