Note that ETOPS goes beyond just engine ratings, it also requires the operator meet all sorts of additional requirements (what if you DO have to divert - can you have the passengers retrieved within 24 hours, do you have food and shelter if you have to divert to a non-commercial airport along the flight path, are your technicians doing proper maintenance, etc).

Good point. Edited my comment to add that :)

For transatlantic flights, two important "backups" are in Iceland and in the Azores. I wonder how often airplanes actually have to detour to these airports...

Happens fairly regularly but for passenger issues rather than technical- medical emergencies and disruptive passengers usually.

Airlines have contracts with companies who provide a sat phone link to doctors who have the flight information and medical facilities at possible diversion airfields. Eg MedAire.

It happens: I was on an London - USA flight that detoured to Iceland some years ago. (Fortunately nothing was seriously broken, burning, etc., but there was a warning indication of some kind that the pilot wanted to have checked.)

The view coming in to Keflavik, with whales swimming in the ocean below, was pretty cool.

Years ago I was on a 747 from SFO to LHR that landed in Iceland - oil pressure problem in an engine. A guy drove out with a pickup truck, climbed into the engine using a ladder in the back of the truck, and started hammering away. After a few hours in the terminal we got back on the plane and flew the rest of the way to LHR. I kept wondering just how qualified he was...

Last I know of is https://en.wikipedia.org/wiki/Air_Transat_Flight_236 running out of fuel.