What do you expect out of creatures that were molded by billions of years of evolution? If their ancestors hadn't been willing to fight over pieces of that rock, whether that willingness came from genetics or culture, they wouldn't be here today. I'm not aware of any good reason to believe that's about to change.
With the environment changing so do the evolutionary advantages that come with the environment.
In that regard, an evolutionary trait that is considered disadvantageous in one environment might be actually advantageous in a different environment.
Case in point: Humans fighting over rocks.
When there were only a few humans around but lots of "good rocks" to fight over, being competitive for the few humans might have been an advantage. Because there had been enough "good rocks" around so everybody could get their own piece of rock if they compete and some can even get more than just one rock.
Now change the ratio between humans and "good rocks", whether that be because there are a lot more humans than there used to be or because there are far fewer "good rocks" around to compete for or simply a combination of both.
In such a scenario competition for the sole ownership these few remaining "good rocks" between big numbers of humans, might be a massive resource waste compared to a more cooperative approach that doesn't pit everybody against each other for a scarce resource.
It may certainly be a massive resource waste compared to an idealized system where everyone chooses to selflessly cooperate. But what is going to happen to the people in your scenario who choose not to compete for control over a slice of the rock when other people continue to compete for control over a slice of the rock?
Well, that's the issue: The people not choosing to compete and instead seeking cooperation will be at an individual disadvantage compared to those that compete, so it's hard to get people to cooperate on a massive scale because it's often a rather selfless act. Why cooperate and share, when you can just be greedy and keep it all for yourself with no obvious disadvantage to yourself?
But this results in kinda a chicken&egg problem: If nobody cooperates then we are wasting resources and effort, but if only some people cooperate they put themselves in a disadvantageous position, compared to the remaining competitors, and will have a hard time pulling trough.
The people not choosing to compete and instead seeking cooperation will be at an individual disadvantage compared to those that compete, so it's hard to get people to cooperate on a massive scale because it's often a rather selfless act.
This is not an insurmountable problem by any means. Group competition occurs on a variety of levels. Nations compete. Ethnic groups compete. Political parties compete. Businesses compete. Individuals compete. When competing as groups, there just need to be benefits shared among the group (i.e. a reason for people to be a part of the group) and ways to punish free-riding (i.e. ways to prevent people from receiving the benefits of the group while not contributing to the group).
The point I'm making, though, is that if any of those entities chooses not to compete with other like entities, it will dwindle into irrelevance or die out. That entity will not be making the world a better place by choosing not to compete. Instead it will simply be replaced by other such entities which are more inclined to put their own interests first, as has happened to countless other nations, ethnic groups, political factions, businesses, and individuals that did not compete successfully for whatever reason.
how it used to be: you fight, win and get some rewards or loose and leave with a bleeding nose. How it is now: you fight and everybody dies from nuclear / bio / nano / .. weapon.
