OP: the best advice anyone can give you at this point is to enjoy what I presume is your vacation and not waste time or energy fighting a Sisyphean battle with AA. You got to your destination and BA resolved the situation for you. It's extremely unlikely you are going to get any significant monetary compensation out of AA for this. It is just plain not gonna happen that you are going to change the service culture or customer service policies of the giant AA corporation. Not. Gonna. Happen.
The second best piece of advice you can take away from this is that, as posted up thread, the best place to make the reroute request is with an airport agent (or even a phone reservation agent) at the point where your itinerary is originally disrupted. In this case JFK. They have the most flexibility and willingness to help at that time. Once you've accepted a reroute, AA considers the problem solved, and more importantly views you as having accepted a solution, and has no reason to make further changes at a downstream connection point. When that connection point involves a change to non-AA operated flight, there's even less incentive for AA to make additional changes. The same could be said of pretty much any airline.
One thing I don't understand is if your wife and other child were already en route to Geneva, in the event that you didn't get there that day, why couldn't they just have continued on the other ~40km to Annecy and checked into the hotel you had booked there? And thereby making use of that hotel you had booked. Why would they be stranded?
Last edited by bse118; Jul 15, 2016 at 9:49 pm
Reason: typo