Publicly? It's one thing to be denial, it's another to acknowledge it to yourself but still elect not to engage in a public display of this. The strongest men I've ever met were not in denial of their emotional state but didn't need to engage in public displays like those listed in the article.
On the one hand yes, there's no need for emotional display for its own sake. As far as I've observed, that is mostly a cultural (or really a subcultural) marker anyhow; some communities value it, that's fine. I'm not personally that comfortable with emotion as a tap that gets turned on and off on demand. (During the group I mentioned above, I told them that if they kept asking me what I was feeling, the only feeling they were likely to get out of me was anger about being expected to feel something.)
That sort of cultural practice is a different thing, though—completely different—from deep feeling coming into experience as part of a healing process. The latter is not a tap and there is no quality of display about it. It is more like a flood that rises from forgotten places, and the healing comes from surrendering to that flood and allowing it to change whatever changes. Maybe after that you become more emotionally expressive, maybe not; I personally think that matters less than people in these communities say it does. But you become more integrated, gain capacity, become less reactive, less conflicted.
The subtle part comes up when you say "Publicly?". Yes and no. I don't think there's any need for ongoing public emotionality. I do think there's a deep reason why healing can't fully happen in isolation. It's because our wounds happen in relationship, usually in early relational experiences, and what is wounded relationally can only heal relationally. Personally I have found that frustrating—I would rather go off somewhere, resolve my issues myself, be a self-healing organism. But I've come to see how this impulse to isolation is actually an aspect of the wound itself, and so not a way to heal. Wounds don't heal themselves—they just repeat until something changes. As far as I can tell, there's no way to avoid connecting with others if you want that to change. But others' mileage may vary, and in any case this connecting needs to happen in safe, closed environments with strong containers, not in random public settings.
The way to overcome pain is to acknowledge and include it instead of denying it. The difference is huge, and most of us need help to get there—personally contactful help, not idea help. I found it pretty liberating to be able to get such help from other men, in a group no less. The point is not to become a gushy feeling-sharer or a sensitive new-age male. It is to no longer be governed by unconscious feelings and the wounds of early experience.
The organizations that practice this work are not super clear about that distinction, which I suspect limits their appeal to many men. I went because of a friend I respect, who in turn went because of a friend he respects. Had either of us only read an article like this, I doubt we'd have been interested. Its subtle ironizing, which approaches belittling ("chastened menfolk", "There, there"), would have turned me off. The photos would have turned me off too, and I've sat in a lot of workshops (though usually with few men and many women).
Edit: an interesting thing to me was the gap between practice and theory. Even though there were parts of the theory behind that work which I didn't necessarily find appealing, I met quite a few men there who struck me as having a kind of integrated masculinity (maybe not the best phrase, but it's hard to find words for these qualities). They seemed strong and open at the same time. I found myself admiring them and wanting to be more like that myself. None of this had much to do with expressing feelings or being emotional; I would use the word presence instead. They maybe even seemed a bit less emotional than most men I meet—more able to take in what is happening without being reactive.
So I would say the theory doesn't work as well as the practice. You barely ever get that! usually it's the other way around.