As much as I obviously detest guns and gun ownership, adding more restrictions wouldn't do a great deal to help reduce suicide rates.
I don't know, research suggests that removing a quick and easy means to commit suicide does reduce the suicide rate. Fot example, people have scoffed suicide nets on bridges because people would just find another method, but that does not appear to be the case. Also, when England changed its gas system so that it wasn't super easy, fast and painless to kill yourself by sticking your head in an oven, the suicide rate dropped noticeably rather than shifting to more difficult methods. Freakonomics (I think it was the podcast) did a really interesting dive into these numbers.
I do get that, and I'm on record as arguing that a home with no gun in it is safer all round than one with. However, in my own home (which obviously has no guns in it) I can count about five alternative quick and easy suicide methods, were I so inclined. I'm sure there are more if I wanted to get creative.
The argument from the pro-gun crowd that "take away guns and the criminals will just find other weapons to kill the same number of people" has never really held much water, but when discussing suicide rates and methods, in this case it rather does.