Well, it looks like Atari saved some money here! I've done another check with the two gamepads from a friend, who also owns a 2600jr (in pristine condition!) which came with them. Same problem - second fb won't work on a 7800.

It looks almost deliberate, but I guess that this is mostly due to a production problem they were having back then - and then no quality control to rectify. Cheap to the end.

@Niku-Sama: I've replaced the plug (and cable) on one gamepad and now it works just fine!