Sounds like you need to file the claim with american since american transported your bag. This needed to be done at the airport baggage office. We had a bag damaged by them (the handle ripped off when the baggage handler lifted it in front of us. We filed the claim and they sent the bag of for repair. They then said it can't be repaired and gave us the option to pick a free bag from a big catalog. We had a samsonite large rolling bag and could have got another samsonite but decided to get a different bag instead.

Also see if the card you booked with has travel insurance. If so you can make a claim with them. Ive had great success with travel insurance paying to replace bags. I usually buy a seperate policy for international. I use generalli