>You can 2v1 from the beginning
But there can only be one winner. To 2v1, the two need a lot of trust. Because what is stopping one of these two from turning on the other when it's profitable, if it's guaranteed that they'll have to compete with eachother sooner or later anyway? To draw a(n unfitting) comparison, it'd be like teaming up in Battle Royale or the Hunger Games: there's only one survivor in the end anyway, so what's stopping you from killing your mates in their sleep? That's four or five competitors gone in a single night with zero effort.
Three person chess would have a psychological element to it as well as a strategic one, as trust becomes important. With normal chess, you can always trust that the enemy wants to win against you.
>not playing four dimensional games
Gretel would beat them all with ease.