I've never really considered there to be a difference between men and women 'cept the naughty bits, we're all just people when it comes to thinking.
That is wishfull thinking, and part of the "humans are a blank slate" belief that was popular in psychology and education during the 80' and 90's and the popular cultural in general, which while well meaning, has proven time and again to be naive and misguided.
Yes, we want to make the world to be as egalitarian as possible, but if you are serious about achieving that effectively, you have to be honnest and realistic about the differences in psychology, culture, socialization, etc that do exist between sexes. Ignoring it is the famous ostrich with their head on a hole strategy. Its more comfortable in the short term, but disastrous in the long one.
Both men and women can benefit from communal man-only/women-only moments. In fact, we naturally create these moments without even noticing it. Women will often arange diner parties, or brunches "for the girls only". Some book clubs are woman only. During family reunions, women will often use the preparation of the food as an excuse for then to share their stories in the kitchen. Equally, men will go on fishing/hunting trips, or go watch a sports match together, play poker or will hang around the grill during the neighborhood barbecue.
I say this as a guy who did not participate in many of these "guy-moments" as a kid, and was more of the artist kid, theater kid, whatever. I didn't see myself as a mainly man, nor did I even want to.
Only in my twenties, I started realizing there were many failings of my character that were directly associated with an underdeveloped "masculine side". Again, I'm using that term as a metaphorical tool. For a dude, I had a more well developed "feminine side" (for dude standards) than my "masculine side". I went on to exercise more, started practicing boxing, hang out with dude-bro guys more often then I did before, without being condescending and arrogant towards them, and it did help enrich my outlook of life and grow as a person significantly. And I am aware my experience is not unlike many other guys my age.
I'd go as far as say that the toxicity of groups such as incels and pick-u0-artists cults is in a way related to that. If you do a cursory survey of the guys in these groups, they are not super-masculine jocks. They are the sensitive/creative boy that is feeling inadequate displaced. They are guys that rarely competed in sports, didn't learn to take risks, to accept defeat graceously when it happens, to work hard on self-improvement and allow that work to be tested against the work of others on the real world (again, I'm describing a lot of my younger self here). And as a result they have a self-loathing, self-pittying and entitled sense of self, and a very judgemental and apocaliptic world view (you are either a Chad or a virgin, and I happen to be a virgin, and therefore I'm a useless embarrassing waste of space, so I might as well off-myself already, and maybe take some innocent people along if only to vent my frustration)
Incels are not an example of toxic masculinity, but of what happens when men DON'T develop their masculinity properly. Its lack of masculinity rather than excess of it.
So I can understand the impetus to seek out this "male spirit" thing and whatnot. The idea itself is not inherently bad, but it depends on the execution.