>The moving company warned us that it was common and they would refuse to move it unless we we willing to forgo any damage claims on those pieces.

I mean, talking about having you over a barrel. They know precisely that you will not stop, and find another mover. All they need to do is stretch wrap the bejeebus out of the bottom to protect those exposed pressed wood chips, but they don't want to have to do that, so they have you give them a waiver and move on to not caring