Weightless objects never make much sense...
Quest items should be droppable/storable/sellable. Just like anything else; if you don't want it, lose it.
But then you get the legions of complainers who'll flood the forums whining about how they can't complete such-and-such quest because they lost such-and-such item, and therefore the game is "broken." Damned if you do and damned if you don't, from the developers' POV.

FWIW, I agree with you. But it seems that we're outnumbered.