I have been trying to join two file geodatabase tables in ArcGIS PRO 2.9.1 and getting an error "ERROR 160333 The table was not found" despite the tool running from within the APRX and the tables are clearly seen.
The solution to this problem was to run the arcpy.management.AddJoin() tool from within the python window after setting arcpy.env.workspace to the geodatabase containing the tables.
It would be useful if you filed the recommended information for the support team as suggested in the link too.
160333: The table was not found.—ArcGIS Pro | Documentation
If you were running the tool from arctoolbox, were you selecting layers with the layer drop-down arrow or navigating to disk? The only environment setting that is honored is the "current workspace"
Tools that honor the Current Workspace environment use the workspace specified as the default location for geoprocessing tool inputs and outputs.
However, it is possible that one of the inputs is not in that workspace and it would really require navigating to that input on disk instead. I can only surmise that it didn't do a good job, since you explicitly set the workspace when running it from python.
Reviewing your method of selecting inputs, table names etc might go a long way to narrowing down where the flaw is.