I should note that the following speech isn’t by me.
Comparing video game violence and hateful speech is not quite right because the former is obvious while the later is not.
I remember when Columbine shooting occurred, many people blamed South Park for inspiring the killers. This is similar to people blaming violent video games for creating violent people. Like you said, those claims are not really supported by the studies. However, is that because people are unable to be influence any media in general, or is it because that particular media cannot does not influence behavior?
I would argue is that what studies show is not that people are never influence by media, but rather that violent media does not influence people to be violent. Most people realize that killing others is wrong. There is little ambiguity about the wrongness about going around killing every prostitute you see (like in GTA). If you were to poll people if machine gunning a public square is acceptable, most would say no. However, what about cases that are not as clear cut? Is it appropriate to call a trans person mentally ill? If you polled that question to people, you would more diverse answers. If you polled the population if it is okay to make fun of someone because they are fat, you would not get a clear cut “no” answer like you would if you asked if being a serial killer is okay.
This is why I think it is incorrect to compare media in general to violent media. Violence is mostly a clear cut unacceptable part of society. No amount of violent games will make people reverse what they obviously already know to be true. However, hateful speech has the opportunity to be more effective because the opinions of hateful speech are more diverse. Maybe you are on the fence as to whether or not you think calling someone a fat slob is okay or not. If you hear it on TV all the time as an acceptable joke, you might find it to be acceptable. Back to my South Park example, that show most probably did not contribute those student shooting up Columbine, or contribute to others committing violent acts. That is because people mostly already know violence is wrong. However, the show probably did increase the amount of people calling other “fatass” or “stupid Jew” or “evil ginger.” That is because those ideas are not obviously wrong for many, so it is possible to influence to lean them in one way or the other.
You can compare this with advertising. No matter how hard the ad companies try, they will never be able to sell me a car. I don’t drive, so I have no need of a car. To me, that ad flies over my head because what I see as the truth is obvious. Similar to how I know that I don’t need a car, I know that there is no condition in which going on a killing spree is fine. However, fast foods ads might work on me. I occasionally eat fast food. The mall I go to has multiple fast food restaurants. Without doubt, my choice to eat at what restaurant on that particular day might be influenced by which ad I most recently saw. The fast food ads would be similar to media with hateful ideas. I am not committed to a certain idea (have no favourite restaurant) so my mind can fluctuate based on what I intake.