In the end, the only choice I had was to do a delete/install of Devonthink (which I had hoped to avoid) - which solved the problem. It’s still unclear why the issue arose, but I had exhausted all other options (including looking at the permissions on the folder the Global Inbox lives in, looking at permissions on the Inbox itself etc.).
So, I’ve fixed the problem, but haven’t solved the crime(!).
Thank you everyone for your input.