Apple Music/iTunes authorization not working

I have a relatively new computer (27-inch iMac, Retina 5k, 2020) on which I am trying to update the music on my iPod from my iTunes library. When I do, I get a message stating that the computer is not authorized. When I try to authorize it from iTunes, nothing happens and the message dialog is displayed again. (I understand that this is due to changes in TLS/security requirements for that program, so no real surprise.)

Going into the Apple Music app, I can authorize the computer, and get a message stating it had previously been authorized and has been re-authorized. However, looking at my account details, the current computer is still not listed, although my previous computer, iPod, and iPad are shown as authorized. Restarting the computer and closing/reopening the Music app does not change this. Looking at my Apple ID/iCloud settings, all 4 devices _are_ shown as associated/authorized devices.

I've followed the instructions in several related discussions, but nothing seems to work and get the new computer listed as an authorized device, and I still can't sync the iPod in iTunes. I _can_ do so with the older computer which _is_ listed as an authorized device, but I don't want to have to rely on that one forever.

Any suggestions on what I need to do to get this resolved? Again, details are:

• 27-inch iMac, Retina 5k, 2020, 40 GB RAM
• MacOS Big Sur 11.7.10
• iPod Touch, 6th generation
• iTunes 12.6.5.3 (used so that I can still have direct management/backup/install for apps, needed for some that are not longer on the App Store)
• Apple Music 1.1.6.37

Thanks for any help you can give!