It might interest you to check out Better Angels of Our Nature, a book by Steven Pinker about the history of violence. He argues that it's typical for humans to fall into cycles of retributive/vengeful violence against one another, or even "Hobbesian trap" patterns of preemptive violence. This leads to astronomically high rates of homicide.

The development of governments that have a monopoly on the use of violence helped curb this. If both sides of a dispute cede their right to violence and entrust a neutral third party with dispensing justice, they can be satisfied with the outcome and save face. For obvious reasons, this leads to a much more civil society than people needing to take the law into their own hands.

In this sense, it's actually a blessing that we're inclined to feel so satisfied seeing justice done by a third party.