So, I think I inadvertently solved my own issue trying to solve another issue.
Settings > Screen Time > Content & Privacy Restrictions > iTunes & App Store Purchases
Make sure "Installing Apps", "Deleting Apps", and "In-app Purchases" all say "Allow." I don't know why this fixed my issue. I originally did this because I was trying to access the App Store to see if the Apple Music App needed an update to fix my issue above. However, I got a message in the App Store saying it was updated, but when I selected the update, it sent me to Safari with the message "Safari could not open the URL because the Link was invalid" message. My research lead me to find the solution I just gave above. I was curious if it somehow fixed my sync issue with iTunes, and it no longer gets stuck on the syncing artwork message. Strange...