You are astute in pointing out that the NRA not only has no substantive relationship with non white people, but also has no intention of establishing one. This can be said of another group:the republican party.
There are stronger, more tangible, more direct, correlations that gun advocacy is linked to the quixotism of white nationalism or supremacism.
The absurdity of this president, the apologists, and sympathizers of gun advocacy to narrowly correlate video games and mental illness as a causation worthy of attribution to gun violence is rife with deception. Such deflection is laughable and mired in all sorts of sophism.
If anything video games offer insight into the plot (good guys versus bad guys) with all its false narratives of inferior races posing a threat to the fear and fragility of the white race — deemed to be honest, good, hardworking, or working class. The dehumanization tactics of this administration plays well into these strange imaginations and hateful concoctions.
This is a form of affective (emotionality) distancing for so-called whites and their placaters for all this violent video games theorizing is merely deflection, just as much is the theorizing of mental illness.
If violent video games are really a form of escapism that in turn desensitizes us from the horrors of actual violence and callousness towards others, then it is simply because whiteness has made this determination by contradicting all the good that violent movies and video games does for capitalism. The billion dollar hollywood movie business has already done much of this desensitizing. Much of that liberal stronghold (so they claim) has come out in protest of the actual racialized escapism and violence taking place with white nationalism.
What this is is just more racialized escapism from their persistent feelings of ill-will and resentment towards other so-called races. Video games, movies, NRA memberships, or Trump rallies can easily be seen as a dimension of escapism. But when such violence happens off-screen because of lax, unfettered gun laws they in effect provide some affective distancing suggesting that the racialized white stress coming from fear, fragility, gripes, resentment and immigration along with their perceptions of aggressive pc culture somewhat fosters this other dimension of escapism.