Why would it? There is no evolutionary pressure to select for "sharing genes". That being said, I don't think greed is the issue. There are plenty of ways to acquire wealth without resorting to war.

No, the real problem with human nature is the tendency of individuals to surrender their individual will to a crowd/mob mentality. That's what allows individuals with malevolent intent to seize power and bewitch the populace, and then go on to do violence. The reason monsters like Hitler or Stalin came to power is not greed, but because others were willing to follow them, were willing to die for them in fact. It is the human ability to cooperate with each other, to put one's own thoughts and ideas and self-interest and the evidence of their senses out of mind while obeying someone else, that is the fatal flaw that leads to so much war. Without it, it would only be the occasional individual fighting another when they can't get along, rather than those individuals having millions of willing followers slaughtering millions of others.