The big issue is that the only spaces like this are either small, private friend groups (e.g. my waterfowl hunting buddies who occasionally get their martial gripes off their chest while out in the duck blind) or public groups/businesses/organizations that inevitably are forced to integrate and become co-ed by outside pressure (defeating the whole point). I disagree with the article's claim that the thing stopping male-only support groups from existing is internal toxicity. I think that external criticism, sometimes unwarranted or malicious criticism from those who have an ideological bone to pick with men's issues or men in general, is a larger barrier.
But we men are also pretty coy about some of these things - if I'm running a men's divorce support group, but I tell everyone it's a golf club? Well of course people are going to assume it's a golf club.