Which is certainly the proper cost/benefit analysis for events *outside* of AA's control/contract; fortunately this case wasn't one of those, and AA clearly owed a hotel room.

It's AA that needed to do the cost/benefit tradeoffs of the system they use to get passengers to those owed rooms -- vouchers, contract rates, etc, are all great, and they were welcome to use them to lower their cost, but since they chose not to (by saving money, not having authorized staff at the airport, not handling it electronically so the phone agent could set it up with the hotel directly, etc), they get to pay the very reasonable cost that the passenger had to do the legwork on. Hiding behind their "contract" rates when the OP *begged* multiple employees to provide such and they refused, is not acceptable.

However, since this is a relatively small amount, I too wouldn't stress and lose sleep over it or sue -- I would submit a complaint to the DOT, who we pay to assist in ensuring that airlines live up to their CoCs and the laws (which do sometimes supersede the contract). AA definitely deserves to have this count against their statistics while getting it to a person who can actually make it right.