Require that everyone get counseling all the time. I don't really have an answer for how "confidential" to keep those records (keeping them 100% confidential is fantastic, but I can also see the merit of knowing your soldier is mentally stable...), but at the very least requiring every single service member at least attend would put them in the room with a therapist and get them accustomed to talking about things.
I'm sure there would still be issues (like a "Has PTSD" checkbox that would cause the same result as you describe), but would it be a net positive?