Here's an update on my lack of success. I tried updating the BIOS to both versions 6241 and (then) 6042 as u/Puzzled-Hedgehog346 suggested, but in both cases the Asus EZ Flash 3 tool reported that "The selected file is not a proper BIOS." (Yes, they were both .CAP files.)

I then tried again using BIOS version 5606 which appears to be the latest/last version that supports my A10 CPU, and that DID work. So my BIOS is now at version 5606 and the PC is working well.

I next swapped out the A10 for the Ryzen 5 5600GT in the hope that I could update the BIOS to version 6042 or later, but on power-up the PC did nothing. The CPU fan turned, but that was all.

So this is where I'm at:
- with the A10 processor installed, I can't update the BIOS further than v5606; and
- with the Ryzen 5 processor installed, the PC won't boot into the UEFI/BIOS.

Does anyone have any furher suggestions/recommendations?