macbook pro – False insufficient-disk-space alert

I am trying to help a senior about 500 miles away who is having serious problems with his MacBook Pro. He lives hours away from any Apple stores and any convenient sources of computer help.

Here’s the situation:
He has a MacBook Pro (Retina, 13-inch, Late 2013) Model A1502 running Big Sur 11.2.3.

He has a 500 GB internal SSD that is slightly less than half full.

The problem began when Mail quit with this alert: <Mail cannot save information about your mailboxes because there isn’t enough space in your home folder. Quit Mail and delete any files you don’t need. Then open Mail again.>

I researched that and found the suggestion to reset the NVRAM and do a Safe Boot. After doing so, Mail was fine for one day and then the situation repeated with the same alert and Mail quits.

While sharing desktops, I noted the red number 4 on the App Store icon in his Dock. We went to the App Store to update four apps and when we tried to do so we got this alert: <Installation failed. Can’t install the software because there is not enough space. 4.81 GB is needed but only 407.8 MB is available>. This is of course non-sensical because he has well over 300 GB of free space on his internal HD.

We tried to download the from Titanium Software and when we did so and dragged the apps icon into the Applications folder, it would not work noting that more space was needed.

And now it gets even more bizarre. I could see that something serious was wrong and wanted to be absolutely sure he had a good backup before proceeding.

I had him attach an empty, 500GB OWC SSD and go to Time Machine preferences, and unlock them, in order to select the attached OWC SSD for use with TM. Even though we could view that OWC SSD icon on his desktop and view it in Disk Utilities, I could not get it to appear in Time Machine preferences so we could initiate a backup.

Using Disk Utility to view that OWC SSD, it listed:
External USB 3.0 Media
Container disk4
OWC Auro Pro X2

I selected each of the above in turn, right clicked on each, ran First Aid, and all passed.

I also looked up Apple Diagnostics Test and told him to restart holding down the D key and note the results, codes, etc. Will post the results here later.

I simply don’t know what to do at this point and really need some help and guidance, for which, many thanks.