I think to identify The Problem, you have to imagine a world where Z doesn't have it, but doing that, you inevitably assign them some "purpose", generally speaking. Like, "lost 40 pounds" or "began enjoying rock music" or something. Or you can go another way about it and think "in passive voice" - imagine the things happening to Z if The Problem was overcome.

Five minutes by the clock give me: 1) in that other world, there's somebody who depends on Z. Maybe he volunteers in an animal shelter, to avoid having to communicate with people too much. There's some reason why breakfast must be eaten, because otherwise, it's not just that life would feel "expectedly worse", but Z's self-image would be "incompetent idiot" - I am not saying he would deserve it, I'm saying this was what I felt when upbraided for eating too little when breast-feeding. I got the impression Z would not have liked this, but without the part where what you do actually matters, it's not going to do any good. So, The Problem would be - Z doesn't have any reason to effect an impact on the world, even what people would call "a burden" might have helped together with medication, for all I know.

2) "things happened independently and made The Problem vanish" make me think of a sudden, inescapable change of circumstances, but I'm having trouble calling up "positives". There seems to be a general resentment towards all those pieces of environment which make Z stay where he is. In that case, I would suggest employment, which would at least give Z a new baseline of 'human idiocy' and some external order to his life. Yes, the world sucks. But there is no real obligation to like it or be liked.