As someone who has taught martial arts to/trained with a fair few military folks, as well as a gamer and one-time indie game developer, I've always thought the link is minimal and not causal.

Convincing the average human to do violence to another takes a _lot_ of effort. Real violence is nothing like games, and has a viscerality that games cannot prepare one for. I would question whether anyone that conflates the two has ever had to inflict injury on another human.

If games made people violent, the military would send people copies of GTA V. Yes, the American army have their own game, but that's a recruitment tool and not a violence-desensitiser.