Migration Assistant can sometimes cause permissions issues (a known and long-standing thing in macOS/OS X). I’d suspect that could have been the issue. But a fresh install of the plugin should definitely clear that up.
I have a new MacBook Pro 14 M1 Pro with the latest version of Monterey (12.0.1) and the latest version of DT3. Followed your instructions to the letter and doesn’t work for me. The plugin that gets installed says ‘Devonthink_BigSur.Mailbundle’. I’ve tried several times. Same result each time. I even rebooted the system several time.
When I upgraded to Mac OS Monterey it turned off Mail’s ability to reveal how many emails were unread in my inbox. I tried everywhere to find how to turn that back on & then I stumbled on “Notifications & Focus” in the “System Preferences”. Choosing Mail on the left & turning “Badge app icon” on brought it back. In case anyone has the same problem.