They seize the opportunity when a smaller group than themselves is separated from the remainder of the big group, and then they overpower the smaller group.

Then they repeat this until they eliminate the bigger group.

My point was that animals understand very well the advantages of belonging to the bigger group at the moment of starting a fight, and they will start the fight only when they estimate that they will win.

The same happens with other social predators. A big group of hyenas will harass or even kill other predators, like leopards or lionesses, which would make them run away when in a small group.

Humans and chimpanzees are probably better at planning a long-term strategy about how to use their advantage in numbers in order to eliminate rivals. Other predators might use their advantage in numbers when an opportunity happens, but they might not perform a surveillance of the actions of a neighbor community, to discover when it becomes possible to use the bulk of their mates against a smaller group of the neighbors that happens to be separated from the others during their foraging activities.