Maybe the average stupid humans (non-politicians) simply need to see a few battling factions, so if all these rational politicians suddenly stopped pretending to disagree with each other, the angry voters would vote for someone genuinely stupid, just to have more variety.

I think it makes more sense to look at the incentives of politicians. Politians want to win. They want to be reelected. That means they have to somehow appear to be better than the other party.

Most politicans also think about their career. They have to impress fellow politicians.