Very possibly reversing causality. That Brookings paper itself states that the economic downturn predates the 'opioid epidemic'. It's based on data that adds up to little more than, well, what you quoted - with little effort to prove which direction the causal link runs, although the paper seems to take for granted it's the drugs causing the labor participation reduction.

The author, describing his assumptions in building his model (pg 36), states "This is a big leap," - language I've never actually seen in an academic paper before. It's usually the sort of thing that peer reviewers say behind their hands before throwing barbs at you.