Actually, in many tabletop systems it's perfectly acceptable for a character's skill to play a role in the damage done by an attack. Someone who is a more accurate shot will have a better chance of hitting a critical spot, or at least closer to center of mass. Which means more damage.
The classic example of this is two hunters in the woods. I live in Michigan, and spent a number of years up in Alaska - many of my friends are long-time hunters, while I've only even held a gun once or twice. So imagine that one of my friends takes me deer hunting.
We're out in the woods and I spot a deer first. Now, I'll be lucky to even hit the deer due to my low "skill level." But even if I do manage to land my shot, there's even less of a chance of it being a "kill shot." I'm much more likely to wound and spook and the deer and then have to track it while it slowly bleeds out. I will have a lower chance of causing enough "hit points" of "damage" to the deer to kill it in one shot. In an ideal system, there's still that chance for a lucky shot but it's much less likely due to my skill level that I'm going to just manage to hit a major artery or organ.
In this hypothetical, however - if my friend happens to be an expert marksman who has been doing this all his life, his shots will due more "damage" due to his skill. He's no longer just trying to hit the deer, he's trying to one of the spots on the deer's body that will result in a clean kill. Even if he's a bit off the mark, his shots will tend towards more critical areas due to his skill and practice. Even a poor shot that still strikes the deer will be more accurate and land closer to a vital organ. So here we see real-world examples of skill doing more damage, or at least trending towards higher tiers of damage. If the deer looks up in that fraction of a second before he pulls the trigger, maybe he just wings it instead - there's still a lot of variables to take into account (which is what the random numbers are meant to account for - all the chaotic variables that come into play.)
For another example - sure, if I'm a very unlucky guy in the real world and I get shot in the face that's going to hurt no matter what. But there's a difference between me getting shot in the ear versus me losing an eye. Both are "head shots," but one will do more "damage" than the other. An expert marksman is going to be more likely to hit a vital part of any location, whereas an unskilled shooter is more likely to land a grazing shot.
In some rules systems, the degree to which you succeed in an attack plays a role in the damage done - so that a skilled marksman is more likely to succeed by a higher margin regardless of the difficulty of the shot, which will result in a higher damage output.
Either way, this is why I'm fine with the newer Fallout games applying character skill and level to damage dealt. From a rules perspective, this makes perfect sense and is nothing out of the ordinary. It actually makes more sense for an attack to deal a specific range of damage that is modified by character skill than for every attack to do the same amount of damage. (Why a weapon doing 1-10 damage is more realistically simulating real-world situations than a weapon that always does 5 damage every time it lands.)