The simple answer is, as you mentioned, the loss of identity. When you can't rebuild an identity...and let's be honest, the majority of veterans are men. We need to work to feel important. I need to work to feel important. If I can't, I don't feel like I'm worth the space that I'm taking up, and that 100% leads to the suicide crisis.
We had the most important job in the world when we were in combat. Then, when you come home and you can't even land a basic job...? I can't imagine what that feels like. I was lucky. I had my education to fall back on when I came home. I fell right back into my teaching career, but a lot don't have that luxury.
To be rejected from jobs because you're either overqualified or.... Nine out of 10 times it's because you're way too qualified, or the other aspect is that I think a lot of Canadian employers don't want to take a risk on a veteran because of the stigma associated with veterans. I don't know how we fix that, but I think you're on the mark with your statement there.