I'm camp hanging, but I think the above statement is debatable. When I search for it, there are too many examples of botched suicide attempts using guns. I have not researched this method extensively, so there might be more nuance to it, but here are my thoughts anyway.
Your comments on "the neural pathways that process pain" are probably correct, but there is no guarantee that a gunshot to the head kills a person. I don't know the exact statistics, but apparently it can happen. No method is perfect, of course, but the problem here is that it's outside of the person's control as far as I know. (Unless there is a technique that ensures effectiveness, which I don't know of.) In contrast, with hanging, the person is much more in control. If the setup doesn't fail and the attempter is not found, the chance of survival is virtually zero. Note that I don't want to go into discussions about what happens if the person is found prematurely – I'm comparing the two methods when both are executed correctly.
The other problem with gunshots is that a person can flinch at the last moment, which will result in the attempter shooting themselves in a way they did not intend. These often result in survival with horrific gunshot wounds. It is caused by SI kicking in at the very last moment and is very difficult to guard against. In comparison, with hanging, there is no such risk. There is a clear threshold (the point of stepping off) before which there is no harm, and after which there is no realistic risk of failure (again, assuming full suspension, executed correctly). SI kicking in at the last moment can't interfere with the method in a way that leads to a disastrous outcome.
I don't know much about guns, but if I were to use one, I would rather use a shotgun that blows half of my head off rather than a pistol. There is still the risk of flinching and missing aim, but at least using a shotgun eliminates the risk of the bullet not killing the person.
Based on these concerns, is there anything about using guns that I don't know?