It's funny, because one time I was having a planned snooze on a 4 engine piston aircraft, when an engine started making an odd noise.* My first officer said 'I don't know how you did that...you were asleep with your mouth open, and suddenly shouted for me to turn onto (Whatever heading).' Even asleep, I knew that the engine had a fault, and that I wanted to get my butt over land. I'm not sure that I would have been so attentive while being absorbed in another thing that I was passionate about.
I think there's a big difference between dozing, in which you are resting and undistracted by anything, and concentrating on some sort of intellectual activity, in which case you are actually diverting your available attention to a different task. A light sleep requires no concentration, so a sudden unfamiliar sound or light can often rouse a person instantly from sleep if he is conditioned to certain sounds but not others. In contrast, concentrating on a video game, a book or manual, or some other activity that requires full attention will inevitably divert attention from other things.
New parents in a noisy apartment understand this well. Traffic may pass all night long outside without waking anyone up … but if the new baby should so much as cough, both parents are immediately awake. Experienced pilots (or operators of any complex machines) are going to be much the same way.
I still think the key is specific "games" designed to maintain alertness without diverting too much attention. These games could emphasize skills that a pilot needs on board, anyway, so that they could also work as training aids. They'd be interesting enough to keep a pilot awake, but "mindless" enough that they would not require any intellectual effort.
I've mentioned Tetris in the past because it's a fair example of this type of game. It is very simple to play and doesn't require any intellectual concentration (unlike, say, chess), but it requires enough to keep you from falling asleep. Thus, you stay awake while playing the game, but any distraction (a cockpit alarm, an engine making a funny sound) is still immediately noticed. I'm sure that more games could be developed that would meet these objectives precisely and help to ensure vigilance without compromising situational awareness.
A good test might be to talk while playing the game. If you cannot talk and play it at the same time, it requires too much of your attention. If you can play it and hold a conversation, it requires enough attention to keep you awake, but not so much that it distracts you.