The only possible advantage could be that he might be able to swing the issue into his being part of a Federally protected class (disabled).

The ADA can't save a job. What it can do is enable you to leave work to get appointments to deal with the problem without getting fired. It's about "reasonable accommodation".

That said, if I got fired for incompetence, I wouldn't request a recommendation from the employer who fired me.

No, but you don't want the employer volunteering something negative when HR from another outfit calls.

An improved mood with meds doesn't equate into improved productivity.

It might equate to better focus when a machine is running or when setting up the machine, which is what I think the core issue is here.

My bottom line question is, once the diagnosis is conclusive (and if I can observe this, the psychiatrist should be able to), should X divulge this to TWC? It might stop a chargeback to the folks who after all employed him for a year, and probably 10 months longer than it should have.