How would it figure out what it means to be "winning"? It might be possible, but would require using OCR to read the 'score' (and guessing as what is meant by the score) and would also need some way to determine if the goal is to "go to the right", etc. (also possible). Doing this would be interesting and more practical in that it could work with games where you do not have access to the memory, but seems like it would require more work in interpreting how the game's state has changed and wouldn't allow going back in time.
A Quake bot is easier to develop because it is written for a specific game.
I accept that bet for training a pigeon to play SMB :), although the scenario you described isn't comparable (it is only one game and you would be giving the pigeon extra feedback about how to play it. It still seems hard and I think it would be notable if you achieved it.)