Every zero-sum adversarial system would be a lot more efficient if they accepted the judgement of a third party instead. How much energy could we save if we dismantled the court system, elections, and pro sports?

None, since players would devote their energy to influencing or becoming the third party instead of playing these games.

DAG coins just distribute the mining work across the entire network including edge nodes. That's not necessarily more efficient but it is more decentralized. Of course it remains to be seen if DAG systems can be robust against attack. I'm familiar with several attacks against DAG-type cryptocurrencies that don't seem to have really good answers.