I finally got it to update. I tried on two different computers.
I even tried resetting my firewall, rebooting my computer, repairing the various Apple components (I run Windows, don't laugh), uninstalling iTunes and all associated Apple components, rebooting, reinstalling iTunes, rebooting, re-authorizing my Apple Music Account, all with no luck.
Tried on a third computer, just for S&Gs, and it worked.
Hmm... I wonder if it's because Apple's servers were getting pounded.
Even iTunes Diagnostics said everything was OK on the first two computers I tried. It synced and backed up just fine.