a nice thought, but i really don't think there is a non mental health affiliated group. the scenarios you mentioned are frustrating, but without some sort of mental health issues, they are very unlikely to lead to serious suicide considerations.
I don't think you can separate mental health from practical considerations though. If you just lost your job, you're flat broke, you've had your utilities turned off and you're about to be evicted, then you're probably going to be at least temporarily depressed. Someone might be thinking of suicide because they're depressed, but the reason for their depression is a practical issue. There is (or at least there was back when I studied psychology) something known as the susceptibility-stress model of mental illness (I forget the exact name). The idea is that it's possible for pretty much anyone to develop a mental illness, under certain circumstances. It's just a matter of what types of mental illness you are more vulnerable to, and how much and what kinds of stress would be necessary for illness to occur. While there isn't a department like the one you described, in the US and Canada, there's 211. If you call 211 or visit 211.org, they can help connect you to healthcare, housing, food, employment, and can help with other things too. What if you're not suicidal though? I want super computers to hook me up with the world's best ass.
The real reason is that it opens a criminal investigation. This will check such things as insurance policies that often become invalid in event of suicide. This helps prevent people killing themselves to solve financial worries. Needless to say the beneficiaries of the policies are also checked to make sure that they did not give the victim a helping hand.
I live in America and I known people who have tried to end their lives and they were put in a hospital so they didn't do again. I have tried a few times in my younger years before I was put on the right medicine. I don't think someone who is truly suicidal should have to be punished enough because they already feel life is too much to want to be around. I read about how young kids are ending their lives because of bullying or Mental Illness. Suicide is sad because its an end to a problem that should be fixed.