You seem to be saying that there are two groups of equal size (people who complete suicide by gun and people who complete suicide by other methods)I think you might be misreading me. The reason it's important that gun-suicide and non-gun-suicide are about the same size is that about equal number of households have guns and don't have guns.
If about 10% of households had access to a gun, and about 10% of suicides were done via gun, I'd say the same thing: access to guns isn't the significant driver of suicide completion.
> But certainly not all that group
Sure, certainly not all. I agree some people would stop. My guess is that around 10% or so.
The problem I have with the thesis that "most people won't switch to another method" is that it leads to a world where the United States is supposed to have an amazingly low suicide rate compared to most other first-world countries. And the thing that is stopping the US is a hot-button political issue, especially because most of the people who are inclined to try and get rid of the guns would disagree that the United States has an especially good mental health system compared to Canada, the UK, Germany, Denmark, Iceland, France, and Finland.
You can point to a bunch of small effects, but you want to apply them to the big picture. Well, why isn't the big picture a lot different? If the US had twice the suicide rate of most other countries, I think you'd be right.