I think jack-of-all-trades characters make a lot of roleplaying sense, in the context of these games. Because this is a universe where most characters adventure alone.
I think specialization works well in a bureaucracy. You have your job to do, I have mine. But if we're out there alone, dependent on ourselves (or, in Skyrim, a companion) for survival, I think we are going to want to cultivate every skill we can learn.
U.S. Frontiersmen and women did this. They learned how to cook, to sew, trap and skin and cure, to fish. They learned how to be stealthy. They learned how to use a rifle, a bow, a knife, a sword, a tree branch, anything they could get their hands on. If magic had been available they would have used that too.
Personally I can't imagine anyone in real life, when attacked, saying, "I'm a pure mage, I won't pick up that iron sword," or "I'm a pure warrior, I refuse to use my magical potential." In real life, facing an enemy alone or with a companion, I think most people would use any skill to survive. In the wilderness, alone or in groups of two or three, someone who is highly specialized is probably going to die.
But there's also the matter of fun. As others have pointed out, gameplay-wise, jacks often become quickly boring. I mostly play specialized characters myself. I do it because it is fun for me, the player, to create interesting limitations for myself. I abandon a bit of roleplaying realism in favor of fun. In my book, having fun is more important than strict, realistic roleplaying.
Those are my thoughts on the subject.