I don't think purely cooperative games really work for chess because it's built for competition, not to mention that most of the fun of cooperative games comes from in-person play, or at least is easier to achieve in-person. I do think more team variants would be interesting, but it's often harder to make them than a standard 2v2 or free-for-all.
Cooperative Varient Ideas

I do think a purely cooperative game where players fight against zombies could be quite fun! Maybe unending waves where you try to survive as long as possible, or one where you all have to fight a single, powerful army. Maybe you could get extra points for surviving the longest, but it's still worth more points to keep others alive with you. Not sure about a ranking system, it might not work for this sort of thing.

This is a very cool idea, and in my opinion, co-ops can be very fun especially if you are playing with people you know. it can really build comradery to be playing on a team with friends.
- one big problem is that there is very little randomness in chess, and players can get very good at it or be really bad, normally skill differences are solved by a matchmaker which pairs better players together, but in a co-op, better players will always win, and poor players will always lose.
- a second problem is that in chess, the position is very bland at the start, but diversity is driven by players competing for objectives (like space, material). Also, even the "best" move is not always actually the best, because if your, and your opponent's playstyles differ, you might chose different moves (for instance, you might give up some advantages to set a trap for an opponent, hoping they will not see the trap) But in a co-op, there will usually be a "best" move in every situation, and you will pretty much always play that move.

Some of the WoFs that we have (such as Circle Challenge) are fully PvE; NCV | Tag Team is Player + Bot vs Player + Bot, so similar concepts have definitely been explored before.

That's an interesting idea, but I feel like the fundamental idea behind chess is to beat your opponent, or the opposing team. It would simply be a different game if it did not involve competition. In short, I think it's a good idea, but it's not chess.
I don’t feel that the competition is as fundamental to chess as the absolute/royal piece being unique and rather short range. However, it probably works better if there is some randomization in a fully cooperative game, as @noahfavelo suggests.
I was thinking that there could be some variants that involve pure cooperation as opposed to only competition, and ones that work for 4 human players. For instance some purely cooperative variants I have in mind are ones for which there's some end position that all players must work together to reach, and ones for which there's a set of rules for qualities the end position must have that all the players must work together to reach. I'm thinking there could maybe be some positions cooperative variants, for which a certain piece from each player is trapped by a few walls, and all the players must work together to get that piece out of the trap.