Just as stealth is incidental to a better story experience, so too is Garrett’s status as a thief. Garrett is a thief not just so players can ‘play as a thief’; he’s also such so that players can receive the game’s story in a certain, very effective way.
The first way Garrett's status as a thief is used in such a way is by establishing an “atmosphere of intimidation.” This atmosphere only exists because Garrett is a thief who must always hide from those around him. Garrett’s surroundings, both the other people and all the buildings, are hostile to him. This gives the character groups and the buildings in the game a “heightened sense”—they tower above Garrett. This would not be the case were players a knight-hero, or something. In this case, players would be on an even plane with those around them. To a heroic knight walking through the City, who does not have to worry about hiding, the appearance, beliefs, style, and overall aura of the Hammerites or other groups in the City would not be as intimidating. Garrett, to the contrary, is skeptical of them and indifferent to them. Everyone is a threat to Garrett.
Garrett also sees the bad side of people. To the average citizen, perhaps the Hammerites would only ever be thought of as good. But Garrett sees them when they think no one is watching, being a thief who can also double as voyeur or snoopster.
The second way story-telling is affected by Garrett’s status as thief is by “Garrett’s disinterest.” Precisely because Garrett is disinterested in the world and the characters around him, their stories become more interesting. They are relayed to the player incidentally. I’ll provide an illustration to demonstrate. If you had a knight-hero in the world of the City who was sent on a quest, by the king let’s say, to either deliver to or pick something up from a Hammerite Temple, learning about the Hammerites would be less interesting. The knight-hero would walk into the Hammerite Temple, freely and with no need to hide, and would read books lying about in the open and carry out dialogue with Hammerite guards. Because the Hammerites would be easily approachable and would be on an even-plane with the player, as they’re a lawful hero, the Hammerites story would appear less interesting. It would not tower above the player.
If you had Garrett, or any like thief, sneaking into a Hammerite Temple to steal valuables, and optionally coming upon journals and other notes lying around—“just-so-happening to”, that is—and the same for overhearing conversations between Hammerites, rather than dialogue with them directly, the story becomes more interesting. This is because Garrett, or the thief, is not supposed to be there and the Hammerites do not know that he is there. Being a fly-on-the-wall is key to Thief more effective method of relaying story.
Along with this, Garrett is only ever interested in his own personal and selfish adventure at any given time. When he goes in to a place, he just wants to steal. Story-consumption is more secondary than it would be for a hero going in to a place on a legitimate quest.
The third way storytelling is made more effective by Garrett’s status as a thief is through the idea of Garrett being “one piece of the puzzle.” Garrett is ‘detached’ piece, if you will, of a much larger puzzle. This would be less the case with a legitimate hero, as one of these would be more attached to the places and people around him, being a legitimate part of their world who does not have to worry about hiding.
Garrett moves by his own will, not by the will of an overarching good force (like, say, a king giving him quests). As Garrett is in his world, the player is by proxy more in their own world, and the game’s story is more interesting because it intersects with or passes by our own world. As a thief like Garrett, players are viewing the world from the perspective of a detached rogue—uninvolved—as if we are viewing it from within a glass globe. The game’s world is more intimidating and alien because of this, and more interesting.