Actually, having a winner and a loser is the definition of a competitve interaction. Without those "classes", it is not a competition. By definition for it to be a competiton, there has to be something you are competing for, food, a mate, a job, a trophy, bragging rights, whatever.
If I am doing some jiyu waza training and I ask my partner to attack strongly and try to counter my techniques, that doesn't mean I am competing with him. If I am sucessful at keeping him off or controlling him, I haven't won anything and he hasn't lost anything. Competition might make it possible to formalize and structure these interactions in some good ways, but it would also change the nature of the interaction, and in my opinion, the nature of that art itself.
Dualisms are dicey. Like the goban or the chessboard -- those are games with two sides in opposition, but not in the sense of competition, exactly -- unless for other stakes, as you suggest. Chess and Go are like war -- one does not really win -- one survives, locally and globally. He who survives with the most, most often, and the longest, gets what's left when we are done.
The play is the thing that occupies our interest, not the result itself, for then the game is over, and if we wanted the game over, why did we decide to play? Delaying a result is part of the game -- mainly for learning how to know at any given moment on which side of the cusp of death or survival you are while the game is still on -- that is always the real lesson in games such as these -- Aikido included.