The usage of other supernatural things present throughout the narrative of the game to say that a mechanic which is never brought up or mentioned throughout this same narrative to me isn't proper support for the existence of Dead Eye as anything more than an in-game mechanic
This is missing the point of that comparison. It was made to show that just because the verse is grounded to some extent in reality, be it its atmosphere, its storyline, or anything else important to that context. It doesn't mean the verse is lacking in supernatural qualities as one of the main arguments made by the OP is that since the verse is grounded to this extent, it doesn't make sense for these characters to have supernatural abilities. It's evidence of the supernatural and a direct counter to the OP's contention, nothing else.
To call it an argument of incredulity misses a big point and that is fact that Red Dead does take care to point out explicit these abnormal/supernatural aspects of its world. Dead Eye being the sole thing it chooses to ignores in this regard, should help illustrate it squarely as a game mechanic
Not inherently so. It's a conclusion that's derived from assuming the lack of statement in regards to Dead Eye equates to it being game mechanics because other supernatural elements are explicitly or implicitly mentioned. It would be improper to assume that the lack of statement deduces the conclusion of it being game mechanics when the premises don't deductively follow. It can be true that even if this statement wasn't made, the ability can still be canonical. Because of this, we must derive conclusion from induction, and so I ask, why exactly would the assumptions you assert have more validity to them compared to the ones made by us?
Nor do I consider the databook telling the player how the mechanic of dead eye functions as sufficient proof it is anything more than a gameplay mechanic either. These are official statements sure, but these statements are not for narrative reasons but gameplay ones.
Well, on its own I would agree that just the databook statement alone isn't enough for it to be considered as non-game mechanics. That's why we're making a Conductive argument, not a Deductive one. It's an argument that each separately supports the conclusion, and becomes more inductively stronger when collectively evaluated. That's why attacking each point separately misses the argument's intention. It's your burden to explain why the premises we've constructed are improper or don't lead to the probability of the conclusion we've reached being true. I don't believe you have sufficed this burden yet.
Guidebooks for games aren't the same as those for books/movies/manga. They exist to both teach of the greater world but also the mechanical workings of a video game. I consider us being far too generous to say that this is a real thing in the verse.
If you concede that it can do both, why exactly can't it be the same case here? If you agree that guidebooks are capable of teaching canonical aspects about the verse, while also providing explanations for gameplay. It can be the case that Dead Eye is an actual ability inverse, and the guidebook is implicitly saying it is by mentioning it while also explaining how the ability works.
I understand you consider such interpretations too generous, but that doesn't really explain your reasoning on why you believe it's too generous. It's just asserting your opinion, unsubstantiated, and that's it.