There are two reasons for playing Devil's advocate. One, its fun. It is fun to challenge the most dominant opinion. Its fun to challenge your own beliefs. And sometimes, especially when you are a person who prides yourself on making good arguments, it is fun to debunk people's arguments. It is a mental exercise - giving those mental muscles a stretch.
But then there is the case when you genuinely want to understand both sides. You want to see where the two (or three or five) sides are coming from because you don't agree with any of them and want to figure out your own stance. Or its just that you want to know more about a situation/theory and don't know where to start. You just want the whole picture. In such situations, there are three stages. First you begin with having an inclination for one side. You listen to their arguments. You are convinced. But then, since you don't want to be biased, you go to the other side. To hear what they have to say. And then somehow you are convinced again. In this stage you are not playing Devil's advocate. You are bouncing from one side to another, really just wishing that you can have both sides debating in one place so you can just observe.
Then there is stage two. You've heard arguments from both sides enough to know when talking to one side what the other side would say. This is the stage when you are not convinced of anything at all. Whatever each side says, you have a counter argument. You don't know what to believe. When you're alone and not talking to any particular side, you might have a gut feeling. But you can't rationalize it because you would get into a cycle of debunking your own arguments.
Then there is the final stage. You become the antithesis of a Devil's advocate. Or its most evolved form. You are listening to one side and you know what the other side is going to say, but you still listen and empathize. In this stage you hold both realities and appreciate them: paralelly, simultaneously. You understand both sides while also understanding that they will never understand each other. You'll begin to accept your gut feel as being your own reality, which is as real as the other realities.
Once in this phase, you will no longer feel the need to play Devil's advocate with both sides. At least not with the people who have their beliefs set. When you meet a fellow wanderer looking for middle ground though, you might stretch those muscles and indulge in some Devil's advocacy. Cause why not? Its fun!