"Which is not to say I'm unsympathetic, but the issue isn't the disparity, it's the things that drive people to suicide."
That's saying that women are incompetent at suicide. It's not like it's a big secret that pills and cutting aren't very likely to actually kill you and getting information of easily accessible methods that will actually get the job done isn't more than a search away (automotive assisted decapitation ftw!). Being capable of researching options isn't a gendered thing (or we should re-evaluate a lot of things).
I suspect the reality is that the disparity is largely based on the rational projections of future life chances. There's a large difference in the likely development of a life for those who aren't completely capable of dealing with it for men and women. Women make an ultimately rational choice to keep chances high to get help, because they have a significant chance of actually getting help, and even women who can never support themselves will often be able to life a somewhat decent life, get support from parents, attract a mate, etc. While men... well, a failed suicide attempt isn't exactly CV improving material.
So, whether a fully conscious choice or not, the disparity is sociologically and probably biologically rational. Men have better reasons to be serious about it if they decide to check out.
And I really don't see any tendencies that it will change. Rather, I think our care for women is biologically hardwired, and the way society is progressing for the moment, being unsympathetic to men is more popular than ever. I mean, fuck, look at something like BLM; even if, in reality, the black men are mainly getting shot due to being male rather than being black, would you try launching a 'mens lives matter' movement? I think not.