Does Time Machine tell you the file can't be backed up? It’s a Monterey beta bug

Public betas sometimes have quirks and this is one of them.

Have you recently seen the message The file "%@" could not be backed up after Time Machine warned you of a failure? You're not alone: many people who have installed the macOS 12 Monterey Public Beta have also received the same error. There's no fix for it; it's likely an error in how Time Machine scans for files.