Problem resolved, but I'm not sure how.
At Apple support's direction I did update my iPad Pro to iPadOS 16.7 from 16.6.8.
Also at their direction, I ran Disk utility on every partition, container, and root drive, a process that took nearly two days, including two overnights.
Then completely shut down both devices, walked away for a while, and then restarted, and walked away for a while again.
Then I plugged the iPad into one of the two Thunderbolt/USB4 ports, and walked away again. This time, finally, the iMac recognized the iPad, though it took at least 2 minutes to do so, and I was able to back it up.
HOWEVER - this morning we were back to ground zero, but with a new, literal, twist. I plugged the USB-C cable in, and nothing happened. On a hunch, I unplugged the cable, rotated the cable, and plugged it back in. Connected immediately. Asked for a password on the iPad. Backed up.
This new behavior is repeating consistently. Plug in one way and it charges but is not recognized as a device by the iMac. Unplug, rotate the connector, and replug, and everything works. Two different cables, same behavior. Both cable behave normally when plugging the same iPad Pro into a 2015 MacBook Pro with USB-C ports.
Go figure.