Skepticism about the claims of instrument variable validity/exclusion through a statistical test—the Arellano-Bond Test
If yes, then how does this square with the general point that
causality/exclusion cannot generally be established with statistical
tests...

It seems to me that &quot;[exogeneity of IV] cannot generally be established with statistical tests&quot; does not imply that it cannot be tested in specific cases. In this (very specific) context, the exogeneity claim rests on absence of serial correlation, which in principle can be tested as a null.

Or, is it merely a diagnostic that may be used as a secondary
argument along with primarily qualitative arguments for exclusion?

I would agree with you there. A non-rejection of the no-serial correlation null is by itself insufficient to establish exogeneity.
(If, hypothetically, the null hypothesis is presence of serial correlation, then yes, but such tests seems statistically unfeasible.)