(I have read your link with additional context)
This stuff is really frustrating, I'm sorry that you're having to deal with this stress. It can feel like we have an obligation to try and talk some sense into people who are on a harmful path, but remember that your first duty to humanity is your duty to look after yourself, to the best of your ability.
The saddest thing is that I do think that cishet men suffer a lot under the patriarchy, and I think that's what can make people like your friend vulnerable to the alt right messaging — they may be aware that they're considered privileged, but without an understanding of systemic oppression means, they interpret that as "they're saying that being a man makes things easy for me. If I'm so privileged, how come I feel so lonely and unfulfilled?". At the core of many of these men is someone who is struggling in some way, and then struggling a bit more because the system we exist in says that they're not allowed to be struggling. My sympathy ends when they internalise the right wing rhetoric though.
The problem with that rhetoric is that it looks at life like it's a zero sum game, where any player gaining "points" does so at the expense of another player's points. They believe that nothing comes for free, and they distrust concepts like altruism, and compassion, because clearly that person is just trying to get a leg up on everyone else. If the idea of systemic oppression is even broached, it becomes a "suffering Olympics", of who has it worse, which is a nonsensical way of looking at things. I'm cis, white and disabled, am I more or less oppressed than a black, cis, able bodied person, or a white, trans person? I don't know, and I don't care, because I do know that each of these people has insights that I don't, so their viewpoint is essential for building an intersectional understanding of oppression, as is mine.
People like your friend often talk as if they are advocating for men's issues, but all they care about is stealing the stage from others. Being cishet is the reason why he's not a part of the LGBTQ community, but it sounds like him being an asshole is contributing to you not wanting him as part of your wider community.