The long one: video games do not turn anyone into anything; maybe, if you like them, they may turn you into a gamer.
There is literature on media fruition showing that media (books, movies, video games) may influence our behavior; it seems that video games are somewhat stronger than other media because of interactive and immersive properties. So, they can certainly help you learn new things, and modify the way you interact with the world. Many video games (certainly not all of them) feature aggressive contents (because they often feature conflict between opposing forces for narrative tension, e.g., good and evil); many of these include violent content too.
Again, the literature shows that violent video games could make you more aggressive in the short term, with no sensible effects on your behavior in the long run; possibly, long-term playing with violent content may make you more aggressive/violent, as well as long-term playing as evil characters (or good ones) may make you less empathic (or more).
Anyway, today we know this is not a simple cause-effect relation: protective/risky factors are in the way and dramatically influence the final result, such as personality, education, social context, personal values, genetics, culture, emotions, mental and physical health, motivation and intention. In the end, all of these factors are more important to explain violent behavior than exposure to media content.
And, in any case: one thing is to be an aggressive person; another thing is to be a violent person; another thing again is to be a mass shooter.
So, to respond to you question, no, video games do not turn people into mass killers, the same way the sunlight does not turn people into piles of ashes.
As I see it, video games do not necessarily lead to antisocial behavior. As I psychologist, I know of no behavior that, say, is provoked by a single factor. Even so, video games often are associated with antisocial behavior. In other words, video games may be a condition, but not a cause, of aggression or other antisocial behaviors.
Of course, this is a short anwer to your question. Note that that video games can be positively-oriented , that is, aimed at fostering positive behaviors, such as prosocial behavior, empathy and the like.
I say NO. In my country the killers never owned video games as adolescents. I happen to come from a family where the Adolescents played a variety of video games from violent ones to adventure, most are grown disciplined and responsible adults
From my point of view, I say no. Nonetheless, according to Ferguson (2007) metha-analysis, violent video games may help players on its biospatial cognition. Spatial visual perception increases related to perceived objects. They are more "sharp". There´s no relation between violent video games and violent behavior.