Join Date: Aug 2011
Location: Barcelona, London, on a plane
Programs: IB+ Gold, TK E+, Hyatt Globalist, Marriott LTP, IHG Diamond
Posts: 14,182
You should almost always do travel insurance before bothering with the airline. Especially when it's third party ground handling who is responsible.
Another reason to approach Amex is that you might find out that its insurance doesn't cover nearly as much as you think it does. Better to find out over a suitcase than a major injury...