Would more men be open to going to therapy if they had resources tailored specifically for them, and if the office had Emotional Support Animals for appointment use?
Therapy is never a cure. If you have a problem then therapy can help you confront it and deal with it, but if we have a massive society-wide mental health crisis it will not be solved by each man individually seeking therapy.
We need meaning in our lives, something worth dedicating ourselves to, and the means to pursue it. Mercenary capitalism won't provide that.
In short, I don't see an answer. I think it will get worse.
Addendum:
Most importantly, this new wave of mental health problems is not caused by a new wave of "not being vulnerable." It's a societal issue and must be confronted there, not shunted onto each individual man.
Even if there were enough therapists, and even if they could actually help, you're still talking about fixing mental problems instead of preventing them.
If these mental problems arise from the living conditions of the masses, no amount of therapy can prevent them. Many that seem incurable might just be failing to escape psyche-wrecking situations.