Like I said, it's complicated. A lot of different factors go into what drive suicide rates and I won't claim to understand them. But having an easy quick way to kill yourself on hand is one of them. Most suicides are impulsive acts with little forethought and the victim will in most cases reconsider if the are delayed long enough to think about it. For examples simple fences on bridges greatly reduces the number of suicides because climbing over them gives you enough time to look down and decided that maybe you don't want to jump after all. And there does seem to be strong evidence supporting that being to case with handguns: if the suicidal person can't just pull a gun out of a drawer and shoot themselves but have to drive to a gun range, fill out paperwork, wait a few days etc they will change their mind.
It wouldn't fix the problem by itself, and isn't a replacement for professional help, but gun control would likely help.