That is true, such spots should exist as well.

But the point of the men-only ones is to allow men to socialize with other men and talk about problems that affect them only or problems they don't feel comfortable talking about in a mixed setting.

No, the two sexes are different and we don't heal the same way or doing the same things.

Would this argument hold up in reverse for a women's shelter?

Men behave differently around women, compared to an all-male environment. It's mostly subconscious, but men (especially the older generation) would instinctively put up a tough front while women are present, and would be hesitant to show signs of weakness, so some would have trouble truly opening up. No doubt there are also male health concerns that men would feel more comfortable talking about to other men.

It's not sexist to want men only spaces, just like women only gyms and salons are not sexist.