I've played 3-handed chess before. We played until the king is captured, ignoring check and checkmate. And we also played so that when you capture someone's king, you gain control of their pieces.
Usually what happens is trading down, maintaining a roughly even 3-way balance of power, until you get to an endgame; in the endgame, someone usually ends up having a pawn that can't be stopped.
We didn't need a points system, we played last man standing; I think that works better in 3-way than 4-way, since in 3-way, if you weaken yourself with a trade, the second-weakest player needs your help against the top dog, so you don't get eliminated, whereas in four-player, if you weaken yourself, you only need three people to maintain a balance of power, so whoever gets weakened first can be safely eliminated.
Sound cool!