Mortal Online had this crazy skill system where every skill raised other skills that were related to it. I know that game is horrible, but I think something similar would make sense.
Firing bolt action rifle X would raise your skill in:
bolt action rifle X's individual skill level by a significant amount
bolt action rifles in general by a moderate amount
riles by a slight amount
firearms by a very small amount
perhaps a tiny amount to weapons of the same caliber or made by the same manufacturer
I personally don't shoot guns, but I can compare it to something I do a lot, which is playing instruments. I have guitars, a ukulele, a keyboard, etc. I can play my acoustic and electric guitar, but they do, in a sense, call for different technique. Practicing one helps me when I play the other, but not everything I learn/know translates. Even less of the knowledge I gain musically transfers over to keyboard, but I still have a headstart over someone who has never played an instrument, because I know about music fundamentals, and so on. I go through phases where I don't play one instrument for a year, pick it up, and still can play it, because skill degradation is not terribly present in real life, but I'll get to that later.
I don't think this game needs a lot of different skills and numbers flying around. You'll naturally get better at firing weapons in-game because you'll be spending a lot of time practicing doing that. The same way that I don't need to slowly and passively become a better shot in Call of Duty. The more I play, the better I get. If I take a break, I might get a little rusty, but I don't forget everything I know, which leads me to my next point:
Skill degradation is annoying. If some kind of skill system IS implemented, I hope I can only forget a certain degree of what I learn. With things like shooting a gun, I imagine it becomes almost instinctive after a while. I know people have "sorry, I haven't done this in a while" moments, but very rarely does someone who used to be adept at a given skill recess to the point where they have absolutely no expertise in it whatsoever, no matter how much time goes by. Obviously with things like fitness, that's something you constantly have to stay on top of, I'm only talking about degradation of skills/expertise.
I think the easiest (and most realistic) way to handle this is just to let your skill meter exist in the real world. You have to remember that you have as much experience firing a virtual weapon as anyone else in the game does when you start, and veterans will be able to make better shots than noobs. It's true for any game. I don't see why this kind of system is needed.
Skill degradation is what happens in real life. If you sit on your sweep all day, you will be less fit. If you're a builder and you don't build anything for a while, you lose some of your skill. I think you're overestimating the amount skills will be decreasing overtime. Don't forget that you won't be shooting a whole lot unless you're in a big party with a lot of resources. Also, a character with a higher shooting skill would probably jam his gun less, have a steadier aim, and know how to fix the gun better if it does jam, for example.
Just one question, how will shooting a shotgun made by Guns, Inc. make me better with Guns, Inc.'s sniper rifle? Maybe the same caliber would make sense, but not same manufacturer.