Disappointing liberal article. Advertising did not make something from nothing; gender disparities in meat consumption are older than that. Check out The Sexual Politics of Meat, which answers this question in great detail. It's complicated, but basically:

1. Women are objectified, reduced to specific aspects of their flesh instead of considered as entire human beings. Tits or ass?
2. Animals are literally objectified; a neatly packaged grocery store steak has no relation to the once-living cow it was cut out of.
3. This "absent referent" allows cultural violence against a group without directly recognizing them. For instance, a number of restaurants have animal logos with (human) female secondary sex characteristics that attempt to seduce the patron into eating them. They may joke about chicken "breasts" or "racks" of ribs, etc. Women describe being treated "like a piece of meat"; boomer humor explicitly compares the bodies of women to those of animals to be consumed.
4. Participating in this process is manly, because a man is someone who dominates others by objectifying, fragmenting, and ultimately consuming them. Refusing to turn animals into flesh, instead recognizing them as moral entities, is linked to a refusal to objectify women, instead treating them as people; this makes you gay or a soyboy or whatever.

Please forgive my clunky explanation; I read the book last year. I highly recommend it: it's an easy read and chock-full of evidence.

can we men find a way to not be shitheads in such a way as to bring about communism

Dunno. There are certainly ways men can be better, by reaching a fuller understanding of "traditional" gender roles (which can be largely understood as superstructure for economic relations between men and women rather). There's bad stuff in there that you and I are doing unknowingly. But men being better is not the same thing as being better men: if we take our blueprint of what not to do from what masculinity currently is, we are pointing in a new direction.

It has been the case that men have changed what it means to be a man within the bounds of masculinity. Dworkin gives an example: for a long time going to war made you manly. Then men refused the Vietnam draft, and suddenly that refusal no longer meant cowardice: instead it was a bold statement of your masculine individual power against the system, your individual moral clarity and strength, blah blah. But it was integrated within the existing ideological framework of male supremacy where we claim transcendence (courage, self-knowledge, achievement, ambition, what you do) and relegate women to immanence (passivity, beauty, what you are).

I suppose that if men get "better" the meaning of masculinity may improve, but I think that "revolutionary masculinity" is going to be subsumed into ordinary cultural masculinity and end up reinforcing it. For instance, if a revolutionary man loves the people, that will be incorporated as meaning that a man is capable of self-sacrifice and transcendent love in the old chivalric way, relegating women's love to passivity. I do not know how we can do these things without reifying the Subject/Other division. Perhaps by aspiring to neuter revolutionary ideals, not revolutionary masculinity, with vigilance that the two don't merge. That would need visibility of women who are revolutionaries and whose traditionally-feminine qualities are meaningful parts of that, not just upholding male supremacy values and showing off "mannish" women who are revolutionaries.

So I guess revolutionary masculinity is probably not possible. I suppose that if women achieve real economic liberation from men, the superstructure will no longer need to justify their oppression and we will have an easier time rescuing gender identities from what they are now.