The idea that playing violent video games makes kids violent is insane. Before the positive arguments, it should be pointed out that the burden of proof is on the proponents of this idea because causation is specifically a change and so it is by definition non-default behavior. I don't have to prove that being a chef doesn't make you want to be a cannibal; it doesn't by default and someone would have a burden to prove that it does.
The first real argument is best explained by an example. Tell me if the following argument is reasonable:
Although of course it has the burden of proof, we can't discount entirely the theory that being a chef makes you want to be a cannibal, because the burden of proof only establishes default positions, it does not prove truth. In fact, I even have some statistics showing that a disproportionate number of chefs are cannibals.
The problem with this argument is that being a chef and being a cannibal are not arbitrary characteristics that we know nothing about. We know what these mean and we know from experience how consciousness functions. Our psychology is not an arbitrary mess; we can be sure that being a chef does not lead to being a cannibal in a way that we cannot determine for sure that X scientific theory is correct.
The positive argument for media violence breeding violence is, "but it shows you violence, and thereby desensitizes you to violence". This argument can be shown to be false by pointing to other examples: no one would argue that watching Land Before Time desensitizes kids to life-threatening danger. That's because even a small child clearly knows the difference between fiction and reality and it is obvious from this and every other example that seeing something in fiction is a totally different experience from seeing it for real and therefore cannot desensitize you to the real experience.
Many video games and movies do show gore which is in some ways similar to the real experience of seeing gore, and that could desensitize a person to the experience of seeing gore, but is that the aspect of violence we should be sensitive to? Is the visceral horror of seeing gore what's actually wrong with violence? No, the thing we should be sensitive to is causing harm to innocent people, which violence in games cannot desensitize a person to because it does not give them the experience. It does not make them think they are experiencing real violence.
This is where I anticipate the "but some people might have a mental illness that gives them difficulty distinguishing reality from games" (this argument was actually raised to me while I was drafting this article, albeit by someone playing devil's advocate). Oh boy there are so many things wrong with this argument. We need a list.
If someone can't distinguish fiction from reality, it's not just violence that's going to affect them. Almost any form of pretending is probably bad for that person.
Such a group of people, if they even exist, are certainly a small minority. Blaming violent video games for an effect on them is like saying peanuts are to blame for the deaths of people who have lethal peanut allergies and either don't know it or eat them anyway. Peanuts should be age-gated or banned altogether!
Finally, this concept of "mental illness" is illegitimate anyway. It's just a FUD tactic to say, "Well maybe there are people whose minds don't work correctly and so no rational principle of human psychology or behavior applies to them". If you want me to believe there are people that despite living in the world for years are incapable of distinguishing it from television despite everything I know and have seen about human beings showing that this learning is inevitable, you're going to have to prove it.
One thing that can and does encourage violence is glorified portrayal. A game showing gore does not make you want to shed it; a story with the message that violence is okay in a situation it isn't - such as because the target broke some arbitrary rules ordained by people he owes to nothing to - can. Contrary to popular saying, almost no games do this outside of the government exception; the enemies they have you fight are most commonly strawman chaotic evil attackers that you must kill to protect innocent people. So most "violent" games are teaching kids that it's okay to kill terrorists, battle droids, Sith lords, or other clearly legitimate targets for violence; not encouraging actual aggression.
Of course the other side is that some games do legitimize violence in the name of the government - but this is only true insofar as most people who make this argument would defend the acts in real life, and also so does every other medium, such as books.
In conclusion, we can understand human psychology without gathering data, violent games and movies do not breed violence, and there are no valid reasons to be suspicious of them as a category; what we should be suspicious of is narratives that legitimize clearly immoral acts.