Why can't I use iTunes to add my iPod nano as an authorized device?

You can’t put songs from Apple Music onto an iPod, as far as I know. The reason for this is because the iPod has no Wi-Fi, and Apple Music is a streaming service with a monthly subscription. If they let people download the songs from Apple Music onto an iPod with no Wi-Fi, people would be able to keep them forever, as the iPod wouldn't know if the person unsubscribed from Apple Music. They could’ve made it so that you have to plug your iPod into your computer every month to make sure that you’re still subscribed, but there’s an option to change date and time on the iPod, so people could cheat the system. Also, Apple Music was created long after Apple stopped updating the old iPods.