Women and children would eat better in many cases with no men around--because gathering is more efficient than hunting. But the society would get wiped out by rivals that had men. Game hunting is then the strategy that gives the most efficient combat training while also reducing the costs of feeding the fighters. It makes sense with the idea that the best hunters are also the most desirable mates, because not only are they directly providing food, they are also demonstrating their proficiency at a skill that is used as a proxy to judge prowess in warfare.
That's my theory, anyway.