The Navy needs a better screening policy. The entering assumption for military service is that the individual is a fit, healthy person. That should include a mental health evaluation. If working 60-80 hours a week is going to make you shoot yourself, this job isn't for you. If being given blunt feedback about subpar performance is going to make you shoot yourself, this job isn't for you. We don't generally don't take people into the military if they'll need chronic care for heart issues, lung issues, whatever, but somehow mental health is the exception.
However, where I will agree with you is that we unduly rely on the clergy for mental health services. I thankfully have never sought out mental health counseling in the Navy, but if I did a priest is the last motherfucker I want to talk to.