I think a large factor aside mentioning males pick more effective methods is the mix of society and biological ideas of honor. Medically speaking men are more disposable, any good parent would give their lives to protect their children, we've all heard horror stories of that with out a doubt it's a mix of love and biology we do so. But if we have a partner males are the protector we hold our partner in high esteem, and fitness (in a species way) hence we are with them. If a husband and wife were crossing the street the man taking the car hit to protect the woman is much more common and reasonable.
I in no way do mean this as women are dishonorable or lesser for it it's just how we are built. I've also over the years just through casual talk found in "what ifs" of dire situations like super cancer, coma etc. The guys all seem to say pull the plug or I'll do it, girls it's a mixed bag. Ask a few guys and girls "what would make you kill yourself?" 99% of the time men will have a longer list if any at all.
I think we also have a hard time with "grey area" suicide, easiest example is historically almost all male of something like a rescue or combat. Running back into the burning building, or not leaving your bunker. Women haven't been exposed to these things as much and the cultural push is a bit more respectful for men who take their own lives even if not in a noble deed. So I can see how someone super depressed of losing a job would feel among those ranks as Leonides by not becoming a burden.
I think very much we as a society need to rethink how we feel and approach those who are at risk and understand what I consider the "real" number of suicides to be incredibly higher than what is listed. A guy who's upset and sees no reason to change his chain smoking, drug use, poor eating etc but holds a job when his liver fails it's cancer who got him, not he was black out drunk 7 days a week for 20 years. If someone got some kind of push to sort himself out early or if he literally felt unfixable and didn't feel pressured out of eating a .45 sundae forcing him to take a harsh slow road, but I don't want to swap topics to if suicide is ok or should be seen as such.
Over all I think it's a mix of biology and culture and even in a system of perfect medical and mental health care, men still would more often.
Little personal evidence, friends I have whom are married, both signed over power for that to me because neither agree, one wants plug pulled other not, both disagree with each other and don't trust families. Guess who's who.