ADA always looks like he's getting hurt sliding into second base, I feel you on that one.
And trust me, playing everyday can be quite the drag, especially when you throw things in like travel, etc. There is a mental/psychological toll that goes along with spending all your waking hours focused on this one little thing. You may not always "feel" the physical benefit that comes with rest, but over time one believes that you are making the players' body a little more resistant to injury.
You also don't want to be in a position where your entire bench is useless because they never see pitching. It is a hard thing to balance, especially in the American League. You don't want to screw with your best lineup but you need to give those bench guys some at bats, since you're bound to need them for both pinch hitting as well as potential long term fill-ins for injury.