While I respect your statement that suicide is a topic to be approached with sensitivity and consideration *and* agree that asking the question whether suicide rates alone closes the case on "how men really are", I don't find your opposition to the core message of the article convincing. Suicide rates in a given community of people can be affected by a number of things, for example a religious belief that it is spiritually unacceptable. Looking at overall mental health, incidence of violence, and broadly applied indicators of health, happiness, and safety is a more meaningful exercise IMO.
I do have to say too, decrying the "feminization" of men I find that to be a poor choice of language. This is a dog-whistle for fear-based politics. There is ample room in masculinity for healthy change without playing to people's worst impulses. (Incidentally, this is doing exactly what the article is warning against...)
I do think that the larger context matters: we have to consider what kind of culture a given community seeks when considering what makes a healthy mindset for its citizens. I would argue that in a progressive, secular democracy that these "traditional" ideas of masculine violence, stoicism, control, hypersexuality, and rigid gender roles are antithetical to healthy lives, just as pointed out by the article and the supporting studies. Maybe that is/was different in an agrarian religious community with proximal threats to the community's survival. But that's not the world we live in today (at least in the countries where this study was completed).