The computer was connected to a Thunderbolt Display during the update which I assume had the same effect of changing the refresh rate to something other-than-ProMotion that the linked article mentions. I had to do a DFU restore from another Mac and then run the macOS Sonoma installer from USB, which thankfully detected the existing install and did an in-place upgrade, preserving all of my data. Nothing else worked.
I also wasted far too much time trying to get the DFU restore to work before discovering that you cannot use a Thunderbolt cable — it has to be done using a plain USB-C cable, otherwise the Apple Configurator simply won't detect the other Mac.