You'll have to go back further than chimpanzees to find a society without property ownership. Once monkeys stop eating all the food in their hands the second it gets there and start saving some for later, who gets to eat saved food becomes an issue. It's moral to own property -- i.e., to prevent another from using something -- because ought implies can, and in most cases it's physically impossible for everybody to get to use the same thing. So the problem to be solved is who should own what, not whether there should be any owning going on. And owning things you don't use is mostly a moot point -- when people and other chimpanzees aren't using something but still exclude others, they're normally saving it for future use -- that's the whole reason there's something to be owned in the first place.
Why can you possess something you don't need? That's another way to ask "Why isn't everything owned by whoever needs it most?". You wrote "Once we decided we could own stuff, this immediately led to disputes and fights." That's backwards. There were always disputes and fights. Chimps owning food another chimp needs more is a way to reduce disputes and fights. There's never going to be any consensus about who needs a piece of food most. It's hard to see how an "Everything is owned by whoever needs it most." rule could possibly evolve. In a population where uneaten food will be grabbed by any monkey who thinks he needs it and a fight will start whenever there are two who think they need it, genes for eating every bit of food as soon as you pull it off a plant will be selected for, and genes for grabbing uneaten food even if you're only slightly hungry will be selected for; but a monkey who picks a fruit and carries it around until a hungrier monkey takes it will be deselected.
What makes you think it isn't ingrained in our DNA? Property rights are a human universal. Chimpanzees cooperate; they respect property rights; and it's hard for cooperation to evolve without property rights. Cooperation, at root, is a food storage technology -- you store the food you don't need right away in another animal's body, and when he has some he returns the favor. Inventing a food storage technology that works and that won't be torpedoed by natural selection for uncooperative genes is not an easy trick to pull off. There are only so many solutions. Wasps have one: a wasp is more closely related to her sister than to her own daughter so she has no incentive not to share. Vampire bats found another -- they store the food in their own bellies, so needier bats can't take it, and when they choose to share they regurgitate the blood to one another. Chimpanzees invented a third solution: property rights.At the risk of repeating myself again, if humans are to do better than chimpanzees, we must cooperate. This is critical in all environments, but once we move out of the tropical forest, where we moved with the ripening fruit, it became absolutely imperative. We could either exist in tight cooperative groups, or we die. The rules we develop to insure cooperation are what we call morality. As our societies and cultures become more complex, especially as we compete with each other for resources, the rules become more complex. There are layers upon layers in our morality. It is so deep and dense, we take it for granted, as if it is ingrained in our DNA. It's not. We must learn these rules and abide by them as we grow.
What makes me think property rights are not ingrained in our DNA. It's probably my unpublished data gathered while raising toddlers.