I recently heard about mathematical paradoxes and I have a perhaps strange question: It seems to me that the goal is to figure out what the fundamental problem is, i.e. what gives rise to the paradox, so we can perhaps rewrite the axioms so that the problems disappear. But why not just say: "Well, paradoxes arise when you talk about sets that contain every set, so let's avoid talk about sets that contain empty sets." (Kind of like saying that bad things happen when you divide something with zero, so don't do it!)
Let me add one other thing. I thought the first thing you said was aboslutely right: "the goal is to figure out what the fundamental problem is, i.e. what gives rise to the paradox". The reason is that it is supposed that our being led to paradox in the case of, say, sets or truth or vagueness shows us that there is something about sets or truth or and vagueness that we don't really understand. If we understood things properly, we would understand how the paradox could be avoided, and not simply because we put our heads in the sand. So paradoxes are manifestations of our lack of understanding, and it is the lack of understanding that we really want to remedy.