This is a multifaceted question that requires a multifaceted answer. I may not be the person to give that answer, but maybe I can help towards that ultimate goal.
For many games, and many stories, there will always be elements that people will not like. Games that allow you to do whatever you want regardless of context can be enjoyable, again, regardless of context. Stories exist to allow us to explore our own natures, whether that is us fighting crime or causing it, we know where we draw the line at. Death and killing are primal to our natures, it's reflected in our fears and sometimes our darkest dreams. Everyone has rage, everyone has, at one point or another, wanted to kill someone, be it an adult or a child. How we react to these thoughts determines who we are as people.
Video games that allow us to act out actions give us a safe, stable scenario in which to explore that which we cannot explore in real life, and books and other media do as well. Video games are unique, in a sense, that we guide the character to our whims, whether for good or evil. You can just as easily find video game cruelty potential in a light-hearted cute game as you can find video game caring potential in even the most depraved of 'murder simulators' (Note, this refers to games built with the express intent of exploring murder and killing in the context of the main character or other characters in detail.)
When it first came out, DOOM was considered by the Moral Guardians (Those who took it upon themselves to decry and have rejected certain media) to be nothing more than a mere murder simulator, or worse in their minds, a means of recruiting gullible children into a satanic cult. (Nevermind the fact that you were literally killing the forces of Hell itself.) However, it has been shown that, since the advent of the first person shooter, crime had actually gone down, most strikingly around the point when DOOM had been released. The catharsis factor inherent in such games thus had a secondary purpose, whether or not the game itself had an excuse plot or not.
Games like GTA have, inherently, both cruelty and caring potential. You can just as easily brutally kill a prostitute as you can stop a thief, even though the story lies heavily in the criminal actions of the player character. In Fable, you could be good or bad depending on your actions, and in Jade Empire, which in this case was an attempt at not having a good/evil dichotomy (Open Palm, Closed Fist) your choices shaped who you were and how you were seen. Sadly, due to design choices in the story and how actions were handled, it became another good/evil dichotomy instead of one focusing on protecting the weak or letting them fend for themselves. Even then, there were choices that were closer to what they had intended. As an added example, the game, Undertale, explored the themes of good/evil in video games and what would, realistically, happen if you chose either path. In the 'good' path, you manage to free everyone thanks to your stalwart pacifism and desire for friendship. In the 'evil' path, you kill (almost) everyone and at the point of no return, you have your control wrested from you, and when you are given one final choice, the main character, predictably, goes against you if you finally say no.
However, in the exploration of these themes, many people get caught up in more trivial issues, such as whether or not games make people more likely to kill. As with any medium, they make you no more likely to kill than a book or movie. If a person is inclined to kill someone, they will do it, regardless of the justification. Humans, after all, are animals, killing is in our blood. Some choose to kill others of their species, some choose to kill other, and in their minds, lesser animals. The justification will change for each person, and not all who kill enjoy it. A soldier in an army can kill a person for different reasons. Maybe they do indeed enjoy killing, maybe they did it to save their life. My brother, for instance, blew the arm off of an insurgent in Afghanistan. He didn't enjoy the feeling, but he was struck with the realization that he had completely changed one person's life with his actions. (That isn't to say he didn't regret it. He did, after all, save other lives in the line of duty. But, that is his personal dilemma, and he will decide how it effects him on his own terms.)
In my own stories, I personally explore these overarching themes because some don't get the proper exposure in media for what they truly are. Murder, rape, everything that I can bring myself to write for the sake of exploring the theme properly. While obviously a bad thing, characters in my stories have encountered characters, especially major ones, who had either been or witnessed rape. How it effected their lives and how it effected their view of the world is the focus of the themes, rather than the act of itself. Some people will never appreciate this, some will enjoy it for all of the wrong reasons, yet it is for the exposure of what that theme is, particularly, especially in this age where rape is often blamed on the victim rather than the offender, that makes it worthwhile, even when I feel no worse than the offender in my story. The exploration of this theme, (And why I even mentioned it in the first place,) is as multifaceted as the question and answer. By exploring the theme, I gain better understanding of it. By exploring the theme, and sharing it, I potentially instill that understanding into those who would read my stories. By reading the story, those who gain better understanding learn to empathize, even if minutely, with victims. Those who see it as a glorification learn nothing and further promote ignorance. Those who gain pleasure from it are as they were before, yet those who would enjoy it, but gain understanding of why their actions are deemed bad, (Because in their minds, they are often in the right,) potentially understand why their actions are deplorable. While this last may seem like a pipe dream, it can and has happened.
Now, I don't expect everyone to have read my answer, again, this was not a simple question, and perhaps I didn't answer some of it. But, that is the beauty of discussions such as these, especially when all answers are there for people to see. Each smaller answer, each larger answer, all become one singular answer, told in many parts. One mind cannot comprehend all things, multiple minds can, and if one mind is opened, even a little bit, by another, understanding can be achieved.
