Modeling society as constantly being at war with itself leaves something to be desired. If nothing else, a good game of tug-of-war involves a whole lot of going nowhere. As another reason, we observe that in societies with healthy political systems, the balance of advantage switches regularly between the left and the right, suggesting that there's a feedback mechanism in play that cares about leveling the playing field.
Enter the balance beam model of politics. In this model, the left and right are envisioned as arms on a gymnast as they walk a balance beam. As they pull in their respective directions, the gymnast's weight shifts from left to right. The key distinction is that the gymnast's goal is to not fall off the beam [1]. From this perspective, the intention of pitting the left against the right isn't as a competition to determine which is stronger, but to provide counter-weights to give the gymnast a sense of stability, and to allow them to shift their weight in whichever direction they need to.
[1]What happens when they do? The system degenerates into a monopolar state.
Some key differences between the model and reality:
Balance beams are ordinarily straight, however political systems are called upon to walk a wavy line; it's usually more useful if one side is dominant to the other at any given time.
Gymnasts never insert their hands into their mouths and try to tear themselves apart. [2]
Like with the tug-of-war, the balance beam maintains the fiction of factional unity within the sides, and the fiction of diametric opposition between them.
[2]A gruesome image, yes, but also how it looks when parties decide they would rather undermine shared political institutions rather than hand power back and forth. Or if they find themselves incapable of closing ranks and coming together in the face of external threat.
The role of the enlightened centrist is to help maintain the balance, and to try to get society-cum-gymnast to move in a forward direction along the balance beam. The main advantage that this perspective brings over a tug-of-war is that it highlights that the political left and right are two parts of one political body, which lends itself to a more integrated understanding.