Honestly, unless the OP is primarily concerned with sanctioned RPGA or Living Campaigns (don't really know if they are the same thing) there is no reason this discussion has to boil down to perfectly replicating movie/book/video game characters in D&D. In fact, I don't think D&D has ever been good at this type of thing. This thread has degenerated into silly fanboyism, which is worse than seeing the ladies fight over the last size 6 at Marshall Fields.
I agree with Mal Malenkirk that swapping weapons for the sake of a player's enjoyment is perfectly fine as long as that swap is equitable. Meaning, unless shortbows have clear advantages over crossbows, there's no reason to adhere to the RAW. Obviously the thief class in 4E illustrates a typical archetype for an archetypal setting. Unless the world is really, really small, there is no reason that a thief type from another culture couldn't be trained in a different kind of weapon.
If shortbows do not have clear advantages over crossbows, then I completely disagree with suggestions of multiclassing or feat purchases for the sake of designing a specific type of character that could be created simply by houserule. These suggestions clearly penalize the player who simply wants to dress their character differently from all the other characters of the same type. Now, if the OP wanted to make Garret Strange, a ninja spellcaster, then his choice must obviously include multi-classing, but cosmetics (shortbow vs. crossbow) can and *should* be accommodated by any reasonable DM for the enjoyment of the game and the enjoyment of a player's experience at hir table.
Now, to the simulationists, I wonder if the default PoL campaign supports a crossbow industry. Crossbows were popular because they didn't require as much training to use, great for sieges (defense and offense), and were easier to mass produce. If the world is going to hell in a handbasket, would it be reasonable to say that bows would regain popularity because they are more suitable tools for survival (hunting), and crossbows would be more rare because PoL campaigns cannot support a large specialized populace (craftsman)?