I never played Goldeneye much, but from what I recall the head to head mode was an added thing and the story mode was the feature. My version of that was an SNES game called Metal Warriors, that had a secondary head to head mode.
The pic doesn't do justice, but it was a split screen (obviously) mech fighting game. The exciting part was you could jump out of your robot if it was too damaged and fly a tiny dude around to find the extra one.
Another example is Tekken 3 had a bowling mode you could unlock, which sounds silly but was really fun. Who else found things like this worthwhile?