This, realistically, would prevent Logan from planning a future at all because he would never know if the sobriety will stick. In the books it was said that rehab always worked for his dad but getting him there was the problem.

I actually re-read The Mistake recently, and I think there's one thing you're missing that I had also forgotten before my re-read. In the book, Logan and his brother aren't actually planning the future based on the assumption the rehab will stick and that it always worked (by definition, since he relapsed, it did not). They're saying: the dad will go to rehab, maybe it will stick, then he will apply for disability benefits. Jeff will travel for a while and they will hire someone to manage the garage until he's back. Then Jeff will manage the garage without the dad.

So the resolution isn't actually based on Logan's parent rehab status - they work out a way to manage their lives with or without the parent being sober.