Benefiting from cooperation doesn’t mean something is or is not zero sum. Take war for example. I can team up with others to form an army and conqueror distant lands. By cooperating, everyone in the army benefits more than they could individually, but the source of those benefits comes from the zero-sum expense of others.

The only source (I can think of) of positive sum benefits is innovation. I can, for example, take iron ore, learn how to smelt steel, and use steel to get more iron ore. Smelting iron is still zero-sum, but learning how to smelt iron is what increases the size of the pie.

Model war as hostility between nations or fractions. The fact that this fraction can be internally consistent does not mean that war represents cooperation.

I feel like if you need to redefine cooperation then your argument is a bit of a stretch.