As to what AA "should" do, I am not going to opine and leave myself open. However, as to what they did for us, that's a data point I can offer. A few years ago, 3 of us were scheduled ORD-LHR-BRU, followed by train to Paris, with the other 3 scheduled to follow 2 days later ORD-LHR-BRU, pick up car and meet us in Paris. Over christmas, 12 day driving trip planned for six people; plenty of room in the car for one bag each to avoid need for too many wasted hours at the laundramat. Huge problem with fog at LHR with flights landing, but none taking off. I managed to snag 3 of the remaining seats on the train to Paris and 2 sons and I made it to Paris (out the cost of the train tix which was significantly higher than the tix I had purchased at home for the brussels to paris train). The rest of the group flew in 2 days later (after being assured by AA that the fog was not going to be a problem), were bussed (via ferry) to rotterdam and arrived in Brussels at 4 in the morning without their luggage. This messed up all of our reservations as we ended up a day behind.
Many frustrating phone calls followed. We were repeatedly told that our luggage had been found and would be delivered to our next destination. When it didn't arrive each day we were told that it had not yet been found. Because AA and BA kept telling us it would be there soon, we didn't want to buy a lot of clothes, both because of the cost (at no time was AA willing to commit to reimburse) and because of the time it takes finding clothes in a foreign city. Unfortunately this meant that we had to shop more than once. If anyone needs advice on where to shop in Perpignan... The 3 suitcases arrived at our home 2 weeks to the day after we did; according to their papers, they had been to Italy-we had not.
As to the first 3 of us, the cost of our train tickets was reimbursed. As to the others, we were reimbursed for their hotel in brussels, all of the phone calls, all of the clothes and the price of the suitcase necessary to get them home. We also received a couple of hundred each for the experience. However, all of the reimbursements were made in flight vouchers; since my husband negotiated with them, I don't know if this could have been avoided.
Whole process took months.
Yes, we could have traveled without checking bags (my preference when it's just me and my husband), but that was only part of the problem and part of my ticket price includes the cost of transporting my luggage (which until AA changes their intl baggage policy is a different situation from domestic baggage). The bigger problem, in my eyes, is the misinformation given out once the problem has occurred - something it sounds like the OP experienced too. I take the risk when I check bags of the airline misplacing the bags; I shouldn't have to take the risk of being given misinformation by AA on literally a daily basis for the rest of my trip.
Good luck.