Do you think it's possible to have such a game with out mortal combat as the central theme and still make a big impact in the gaming world. Cooperative but not cooperating to kill the other team?

Worth noting is that "kill the other team" is a blanket term that encompasses many different levels of violence. Yes, in some games you do actually kill people, but there are so many variations - in platformers you often just jump on their cartoon heads, in strategy games it's more about conquest, like playing Risk.

Looking through my own gaming library, there are a lot of completely non-violent games where you could class it as "mortal combat" - for example in Osmos you play a single-celled organism (sort of) trying to survive by absorbing those smaller than you and avoiding those bigger than you.

There are many games out there less violent than Call of Duty that kids could play to bond or make friends, but they get overlooked because they get lumped into a category, and then the media parade the most extreme examples from that category in front of us.

Sports games. I had heard that the killing aspect could have been been removed from the game in Germany for legal reasons. Bullets were to be replaced with paintballs and killing with simply eliminating opponents. They never ended up passing the law banning violent games though.

Both my kids spent considerable time "role playing" and focusing on non-combative activities like chatting and trying out different variations of character choices so I'd say the answer is definitely yes, especially if the game builds on age-appropriate social curiosity or borrows from familiar cultural elements. My daughter especially loved things like playing "barbie" by showing us different combinations of clothes she obtained and playing pranks like walking around cities dressed as an NPC, hoping to fool other players with emotes.