Well, thanks to this comment I went back and had a look at the website the article was published in. The article is already pretty old (from 2015) and the author didn’t publish a lot in the subsequent years. But I found this much more in depth analysis by a sociology professor re-published on this website. Below is a statement by the original author of the article posted here. And yes, this guy seems to have a good grasp on how to deal with the underlying problem of masculinity:

What I have found to be true is that as we access our compassion and put into practice our altruistic caring for women and girls, we collide with our male privilege. If our primary motivation is self-interest, we will not relinquish those privileges and the ongoing benefits we receive due to “toxic” masculinity. We will retreat, internally become silent, talk well, but not change significant behaviors, both personally and institutionally. To me, our willingness to give up our sexist privilege/benefits—including our silence—is the foundation of change, not our immediate self-interest.

If we care about women and children’s lives, we will begin to relinquish those benefits. We will use our remaining male privilege and influence (which we cannot totally discard because of sexist social norms) to undermine patriarchal structures of oppression. We will work to end the violence, harassment, discrimination, income inequality, exploitation, subordination, and danger that women and girls live with every day.

I missed this point in the original article, although yeah, I now better understand why people can still take something from it :)