Is that our responsibility, to save a man from mental illness? I don't feel it is.
I'm offended by the thought and the idea that we should just ignore our ill. What if the person suffering from the mental illness was your brother or sister? Think about if it was you, and no-one reached out to you - how would you feel?
If a person does not go out of their way to do the things you listed, I do not consider that a moral failing on their part. Especially if the cost to them is more than the benefit to the other person.
If you keep jumping on bullets, sooner or later you run out of flesh.
It should be blatantly obvious that, were the situation different, I'd feel differently.
Not everything is a moral crusade, Internet warrior.