You're more awake than I, Jocelyn - I'd already read how Time Machine required links that were incompatible with APFS but still enthusiastically responded to a piece encouraging 'convert all external drives to AFPS too'!
I looked through pretty much everything Google served up, including the Developer forums, for a solution, but found none. Lots of journalists and 'tech gurus' trotting out the official advice that my I knew already (usually in badly written, less clear versions splattered with garish ads) but nothing to help with this outlier / corner case. I suppose it will take time before robust knowledge to deal with exceptions emerges into the public domain - meanwhile, lots of motherhood and apple-pie.
The reformat was not so simple. I used Disk Utility to delete the APFS container after an initial erase left only APFS format options. The drive would not then remount, even with several reboots. It did not show at all in the Disk Utility interface but I could see it in Terminal with the following command (so knew it was not toast):
diskutil apfs list
The disk eventually did appear as I navigated around Disk Utility (I possibly hit Mount but as the drive wasn't available to be selected in the interface I'm not sure how that helped so forgive me for being vague) and I reformatted using Mac OS Extended (Journaled). The drive then offered itself for Time Machine.
The backup did not take long but it is still encrypting (1TB drive, 20 hours later). When running Time Machine afresh on a new Mac, and adding to other backups on the same drive, the encryption was taken care of as the backup completed. I guess this time the disk required further preparation after its ill-fated foray into APFS (which makes sense) and I suspect, although cannot quite recall, that when I did my very first Time Machine backup on that drive some years ago it also took a long time.
Re 'backup backup drive', etc. and, as John said, having more than one bodyguard —
- my Documents folder is replicated real-time to iCloud, which recently and without a glitch restored all saved data to two new machines – good if someone steals my laptop;
- my entire Home folder is backed up incrementally every hour to Amazon AWS S3 Glacier using Arq – helpful if I delete / damage stuff inadvertently myself or want to find older documents;
- and now I again have a working Time Machine backup – for emergency whole-system restores, but I doubt I shall need to rely on this much in the future.