I don't object where the rationality of GG comes from, I am arguing against the notion that they have rationality at all, or atleast a functional one. Even the example you gave, that's rationale for them being apprehensive at best, but any halfway thorough examination of Anita's video's provides that they are not an attack, or that they would be an effective one even if they were. And if that is the case, there have been thousands of attacks on gaming they have failed to even try to stop before GG ever became a thing. They wouldn't argue that they were being immoral at the time, because it is insane to think that gaming should immune to any criticism.
A rationality has to remain consistent to be with facts and reason to be rationality at all. Rationality implies the conformity of one's beliefs with one's reasons to believe, or of one's actions with one's reasons for action.
Gamergate is not rational. There have been numerous instances where their actions contradicted their stated beliefs. It would be like if I wanted to protect my dog. Someone tried to poison him before. So, as a result, my rational is that my dog should not eat any food so that he won't be poisoned. In duty to protect my dog, I have intentionally killed him because I do not associate myself with the fact that he needs to eat. Because my motivation is so rationally incongruent with my action, I cannot said to have been Kantian moral. Or else, if you're going to dilute rationality to just "X's perspective" however inconsistent it is, then I fail to differentiate it between moral relativity. At this point, X is just doing whatever the hell he wants, and it's always going to be moral because reason and rationality have no place in justifying his rationale.
In your analogy to the dog, you lose something by applying the method you describe. To a GGer, they lose nothing in applying the same logic, as they see little issue with the status quo of the industry. So applying Kantian morality gives them a "nothing to lose" war of ideals to fight.
The Kantian logic comes in when you look at what they propose they are about, even if further examination indicates it's not: ethics in game journalism.
Anita Sarkeesian was not the problem per se, it was that the press began to promote her success and "saw" a media conspiracy intent on completely changing the narrative of video games yet again to a negative. Her, Zoe Quinn, Leigh Alexander and every other woman who was being "signal boosted" by the press were the problem.
So they either believed that they were influencing/coercing the media into shoehorning "SJW" topics into the public eye as detrimental (especially the story from them regarding Quinn) or they became the identifiable talking head for this false perception (as with Anita Sarkeesian). Because the last time that all the press could talk about was someone who was repeatedly saying bad things about games was... *drumroll* Jack Fucking Thompson, and he was a master at getting the press to talk about him. So ladies like Quinn and Sarkeesian became the "poster girls" for manipulation of the press and a "falsified" talking point against video games.
Which is why, in their attempts to silence the dissent, they manipulated it into being a media discussion, an attempt to kill two birds with one stone and expose the media for its "SJW" slant that must have been coerced out of them while also ridding themselves of the ones who made it happen. And while I fully understand that there's a flaw in the logic there, because it automatically supposes that talk of equality, tropes, etc. is due to manipulation and coercion, the press wasn't exactly slim pickings when it came to stories of how their opinions have been questionably manipulated before, so the story became just believable enough that a bandwagon was built and people could find a rational reason to jump on it.
Whether their rationality is bound in fact is not even relevant. To discuss fact in rationality is folly, since modern neuroscience has rendered philosophers' ideas of rationality mostly moot in that we can scientifically prove that no form of rationality occurring from the human mind is dispassionate and always slants due to emotions on the subjects we attempt to apply rational thought to. So clinging to the works of Grayling when discussing rationality is a foolish errand, especially when applied in the same breath as morality. With Kant, the categorical imperative only requires an autonomous will, which can divorce itself from reason and fact at any moment, leaving only a corrupt rationale.
Also, Kantian morality intersects unfavourably with the work of Descartes, in that they are certain because they cast no doubts on what they perceive to be true (as Descartes says several people do to their own detriment) and therefore, their thoughts can be applied "rationally", but only to themselves and those with like-minded "rational" thoughts on the subject.
