Time Machine Backup Hang - Locating Corrupted File

I am convinced that my issue is a corrupted or un-backup-able file.

Yes that is rare, but it is possible. By "rare" I have seen it occur exactly once. It happened to be a corrupted game file which TM simply refused to back up, causing it to become stuck. Once I deleted it TM began working again. The drive itself was and remains perfectly OK.

However, drive corruption can also cause TM to "hang" in the same manner, which can be difficult to correlate because the only way to definitively rule out that possibility is to replace the drive. Erasing and restoring might offer an interim solution, but it is not likely to be permanent fix. You'll know when the same problem reoccurs with increasing frequency. Given your MBA uses flash memory I think a corrupted file is more likely—as you suspect.

Using a different backup drive will also be beneficial in isolating the cause to either the source or backup volume. One and only one Time Machine backup isn't a sufficiently robust backup strategy anyway, so if you haven't done so already buy another one at your earliest convenience and add it to TM's rotation. If it also "hangs" you have effectively isolated the cause to the source.

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To extract potentially relevant Time Machine activity from log consider using the following shell script:

clear; printf '\e[3J' && log show --predicate 'subsystem == "com.apple.TimeMachine"' --info --last 24h | grep -F 'eMac' | grep -Fv 'etat' | awk -F']' '{print substr($0,1,19), $NF}'

Copy (triple-click to select the entire line) and Paste that line in a Terminal window.

The Terminal app is in your Mac's Utilities folder.

It extracts Time Machine activity logged during the previous 24 hours. To change that time period change that value. If Time Machine is running there is no need to interrupt TM to use it. Be advised that log is fairly resource-intensive, and if you are using a portable Mac it will consume a lot of battery power as it runs.

log will need a few moments to extract the Time Machine log data. Wait for it to finish. It is normal for its results to include various "errors" and "failures" and none of them are necessarily an indication of anything wrong.

Copy (Edit > Select All and then Copy) and Paste that Terminal window's contents in a reply to this Discussion. Please omit or obscure any information that you may consider personal.

Quit the Terminal app when you're finished with it.