macOS 10.15 Catalina and later have new apps that replace iTunes, one of which is called Music, leading to potential confusion with the Music subscription service called Apple Music. See What happened to iTunes? - Apple Support. If you are using one of these versions of macOS then syncing operations are managed in Finder, which is the Mac equivalent of Windows Explorer. The Apple Music subscription service lets you add content to your cloud music library which is only able to play in approved apps and devices. iPod classic is not able to use Apple Music content. You also cannot sync content that is located in the cloud.
There is no such thing as a new iPod classic, production stopped over 8 years ago. See Corrupt iPod Classic - Apple Community. What are your HDD stats?
See also Sync your iPhone, iPad, or iPod using your computer - Apple Support. You may need to adjust your sync selections such that new downloads to your computer would be automatically included in the next sync.
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