You might be surprised. I've seen the effectiveness of medieval quilted padding against an arrow from a longbow (said arrow went straight through steel plate but failed to penetrate the cotton quilting), so the idea of using cotton to reduce the effectiveness of fragmentation damage is not necessarily as silly as it might sound.
No I understand how effective it can be. Its rather an interest of mine. For 'efficient' I mean the hauling, storing and management of large bales of the stuff which would have become vermin infested and subject to damp/mold/rot very quickly if exposed. Considering there were easier methods.
Wool has been used as a part of armour on occasions too for much the same reasons as you mentioned.