Newcomb's problem is applicable to the general class of game-type problems where the other players try to guess your actions. As far as I can tell, the only reason to introduce Omega is to avoid having to deal with messy, complicated probability estimates from the other players.

Unfortunately, in a forum where the idea that Omega could actually exist is widely accepted, people get caught up in trying to predict Omega's actions instead of focusing on the problem of decision-making under prediction.