I'd rather stay agnostic on whether or not this is the case; we have very little reliable data on culture under non-marginal paleolithic conditions. I haven't heard of any conclusive skeletal evidence for war in that era (murder yes, war no), but this isn't my field so I could easily be missing some.

This paper argues that coalitionary killings were rare among hunger-gatherer societies, and that warfare as we currently understand it did not come into exitence until the rise of agriculture and sedantism, because prior to those develpments, the average hunter-gatherer band simply didn't have enough accumulated material wealth to make the benefit of raids into another band's territory outweigh the risk of getting ambushed by unseen defenders with projectile weapons who know the territory better than you.

However, the development of the throwing spear, used in conjunction with ambush hunting techniques, ushered in an era in which the enhanced lethality of weaponry amplified the costs of assessment errors, and the necessity of movement also placed intruders at a comparative disadvantage with respect to both detection and assessment. Moreover, asymmetrical detection rather than a numerical imbalance of power determined the outcome of hostile encounters. This reconfiguration of the decisive factors in lethal conflict not only raised the stakes (or potential costs) for would-be aggressors but also rendered the benefit of intercommunity dominance unattainable. Because superior numbers were not invariably decisive in encounters between hunting parties, an initial success would neither materially reduce the stakes for aggressors in subsequent attacks nor make it possible to freely encroach on the territory of a neighboring group that had sustained a casualty. Under these circumstances, aggression resulted in stalemate and a condition analogous to a war of attrition rather than territorial gain.

These developments marked a major turning point in the evolution of lethal intergroup violence and in the character of interrelations between neighboring groups. Although fitness continued to be related to territory size (for food-limited populations in occupied environments), selective circumstances no longer favored aggression as a means of achieving territorial gain. [...] In other words, the development of the throwing spear altered the means of production as well as the social relations of production, distribution, and consumption within groups in fundamental ways that also transformed intergroup relations and influenced subsequent hominid evolution.

As I said, it's a minor point. I'm pretty sure the grandparent's argument will stand or fall the same way regardless.

Well, it's not conclusive evidence by any means, but I did note that we have no hominoid relatives; they're all extinct with a capital E. To me, that implies something more than just us being better at hunting-gathering.

And if we, as a species, did exterminate one or more other hominid species, then it seems a small leap of logic to conclude we did the same to each other whenever similar circumstances came up.