If you look at WCK as you would any other physical acitivity/sport, that might help you in deciding whether or not weight and/or strength training would be useful for you.
My view is that weight/strength training can be problematic for two main reasons. First, pure muscular strength does not play a large role in what we do in the sense we aren't trying to out-muscle our opponent and our power comes from specific motor programs. So the body mechanics, technique, etc. of WCK is not strength-based. From that perspective, developing more "muscle" won't help your body structure, it won't help your technique, it won't help your timing, it won't help your feel for the game, etc. In other words, it won't help you develop those skills that are a part of what we do. Just as strength/weight training won't directly improve your your tennis game or your basketball game or your boxing.
That said, WCK is an athletic activity, and the vehicle for that activity is your body. A body in poor shape and poor condition is not a body that can perform any intense physical activity well. The develoment of your skill in WCK will depend upon and be limited by the athletic condition of your body. To become good at tennis or basketball or boxing or WCK requires that you be in generally good physical shape and condition; to be really good requires that you be in really good shape. If you're not, and your gym, kwoon, school, whatever doesn't focus on that aspect, you need to do it yourself (and it might help to get a good fitness coach or personal trainer).
Second, WCK is a very specific sort of athletic activity and puts specific demands on our body. Often people weight/strength training ignore that aspect -- sometimes because they fall in love with that aspect itself and no longer see it as a means to an end, and sometimes because they just don't understand the underlying demand/mechanism -- and train in ways that may interfere with their WCK development. This is why a good coach/personal trainer can be really helpful.