The argument does not work in reverse. If they had more guns, there would probably be more suicides. End of story.
I am not suggesting that no guns == no suicides, because that would be ridiculous. I am refuting the suggestion that having a gun does not increase your risk of dying by suicide. I submit that having a gun nearby does increase your risk of dying by suicide with ample supporting evidence listed above. I am happy to read studies to the contrary, but it seems pretty clear to me at this point.
I don't think anyone is disputing that having a gun increases the likelihood of succeeding. I just disagree with the idea that the gun contributed to the attempt.
Most of what you just stated I have no disagreements with, but it is worded differently than what has been stated by others before.