Both airlines will do pretty much the same things but it depends at what point one asks. European airlines don’t allow their call centers staff to rebook onto other carriers while North American ones do. North American Airlines also train their gate and lounge agents at ticketing, which European airlines do not do. Quite why I don’t know, but the ultimate result is the same - passengers do get to their destinations. However, the things that I heard from AA gate and telephone agents are unbelievable. AA must ‘empower’ them to fob passengers off if they routinely do that.

Regardless of what different types of agents do, what happened here was simply that the agent followed the rules set by the airline. Misconnect from a non-OW airline or oversleep a flight and AA’s agents will be as powerless as BA’s are when it comes to misconnecting on separate tickets (although I know of many stories where BA did help but passengers asked at the airport rather than over the phone). I am not sure why it is great customer service when you receive exactly what the airline promises you will.