It weighs nothing and it's kept in your inventory because they're possibly the most important single items in the game at this point, and losing it would really svck. So you're forced to keep it so you can't mess up and lose it. If they let players move it, you'd see a forum full of people raging that they lost it and now they can't use it.
I think the real point here is that it's a ridiculous concept to be forced to carry around items like this; especially when the quest is over, you don't need it anymore. All it does is invite more bugs where the items are permanently stuck on you. Morrowind made you responsible for your quest items, I don't see what's so wrong with that concept. You lose it, it's your own fault.