Indeed. You only get burned once

.. and then apparently once again a decade later when you let your guard down

Can Borg do this? Like, something you can set up to run once a month or so?

Borg has a command ("borg check") to test backup metadata. I'm not sure exactly what it does, but it is pretty slow, about 80 minutes to check a 1.6TB backup depending on how busy the backup server is. Another approach might be to mount the backup as a FUSE filesystem and monitor it with something like tripwire. I just thought of that a minute ago and haven't looked into it, so idk if it is workable. At the end of the day you have to occasionally do a full restore to another system, and test everything.