There's not a lot of evidence for war, but there's not a lot of evidence for anything. Very little survives over time. So I don't think a lack of evidence is meaningful yet.
Prior to 500 years ago there was little modern contact with the Americas, Australia, parts of sub-Saharan Africa etc., so observation of lack of conflict in hunter-gatherer bands there is more evidence. Obviously I mean hunter-gatherers, not the Aztec warriors.
I don't see what chimpanzees have to do with humans, unless chimpanzees are sculpting Venus Madonnas and drawing buffalo on caves.
I think this myth comes from early studies, who wanted to see this, and observed no warfare during the 6 months they spent around 20 people. Later studies asked these 20 to reconstruct their family trees & tabulate how each great-uncle died, etc.
(The other possibility is definitional. If we define war as involving thousands of soldiers, then obviously this never happened until societies which could co-ordinate thousands of people arose. May as well define it as involving uniforms & automatic rifles, too.)