I'll take a whack at this, because it's something I've been thinking and journaling a lot about lately.
I think that a lot of masculine traits appear not as a dichotomy of either "toxic" or "nontoxic," but rather on a spectrum, which I've been exploring as "virtuous" or "vicious" ("vicious" in the sense of "being a vice," not "deliberately cruel"). That is, many of the traits that we (as a society) view as masculine (and this is separate from the question of whether these are
intrinsic
to men, say, carried on the Y chromosome or whatever) can be healthy or unhealthy, depending on the degree to which they're taken.
So, for example, self-reliance: I think we can all agree that this can be a healthy and positive thing, and that a man who is self-motivated and able to roll with the punches life throws is sitting toward the "virtuous masculinity" end of the spectrum. On the other hand, self-reliance and stoicism taken too far results in the social isolation that leads to emotional withdrawal or illiteracy, untreated depression, substance abuse, reluctance to utilize health and mental health resources, suicide - much more on the "vicious" side.
With regard to mass shootings, I think there's a comparable effect at work. The masculine trait we can assign to this is (something we might call something like) "agency." On the virtuous end, this means a man who owns his say in what happens in his life, who stands up for what he believes, who stands behind the people he cares about, and who takes action when necessary. On the vicious end, this trait turns into not just a desire, but in fact a need (most likely subconscious), to control - to control not just the circumstances of his own life, but also the behavior of those around him.
So, as to your question: I think that where we start is by analyzing masculine behavior, and identifying these traits that may be healthy in one instance, but unhealthy ("toxic," as used in the literature) when taken to an extreme in another. I think that's the value in articles like these, that attempt to distill out the gender issue that forms a major root of some of these issues. Mass shootings almost certainly can't be tied just to gun culture or mental health; American women are brought up in gun culture as surely as men, after all, and deal with mental health issues as well. And the article makes a good point that it's likely not ideology, either, or we wouldn't see such similarities between the alt-right's obsession with cuckolding (actual or "cultural"), white separatists, or Islamic extremism. What those all have in common, at a meta level, is a fear of a lack of control - male agency taken to the vicious extreme.
As
/u/Zenga99
points out, a likely good step toward reducing toxic expressions of masculinity is promoting masculine traits that tend toward the virtuous end of that spectrum, and demoting the vicious ones. But first, we have to know what we're discussing.