IT seems like a horrible choice; if they were paper documents left at a desk would IT take them over or the unit supervisor/manager? Almost certainly the latter. IT taking over business area responsibilities wherever a computer is involved rather than facilitating business units performing their work is almost always the wrong answer.
The comparison doesn't hold. Paper documents can be found by anyone, they get "open access" automatically, you leave them in your team's area, and the accidental losing scales differently (if you "lose" 1GB of paper documents, you notice).
> rather than facilitating business units performing their work
Sure, and the team which enables business units to deal with this internally is... IT. I'm not saying they have to deal with every single document, but if they enable this, they deal with things that fall through cracks.