In our former house we had a half stairway from the garage down to the basement ... so to get into the house in the winter you would walk to the front of the garage (the wall shared with the house) and then down four steps into the house. That corner tended to be dark so I wired a light fixture into the wiring of the garage door opener light so the light over the stairs would come on when the garage door was opened and stay on as long as the garage door light stayed on which gave me enough time to walk into the house.
I think it has been mentioned but since it seems like the problem is the socket and you don't think it is replaceable you could just wire in a separate light fixture next to the opener. Easy peasy.
If something cannot endure laughter.... it cannot endure.
Patience is the art of concealing your impatience.