The primary drivers of why people choose to attempt suicide are related to anomie and other social, economic, and cultural factors. Which is what sound research, not begun, as is most of Hemenway's, with the premise that lawful gun ownership is a social ill and generalizable risk and thus the methodology used must be designed to support that thesis, consistently finds.

That a given suicidal person, driven to that point by those risk factors, will choose the most lethal means available to them is almost tautological. As any unbiased comparison of our overall suicide rates with those of other developed, yet more gun restrictive, nations shows.

On a practical side, 70%+ of firearm suicides in the US are adult, non-prohibited, male gun owners (conversely, the rate for teens (under 20) all means is <3 per 100k). Competent, autonomous adults can't in practice be affected by any proposed firearm type nor access restrictions, so I'm not sure what policy changes would help the big picture.