This doesn't quite answer the question; I'm having the same issue with the same configuration. The iPod Classic shows up in the Finder window as an external hard drive, and there is no "Sync" button as there is with my iPhone. There is also no way to make it show up in Apple Music.
Last time I synced the iPod was before Christmas. I have 3 household devices that interface with it, and it was handy for Christmas music. There was an error, but the iPod synced anyway. It has done that a few times in 2021, but mostly I use my iPhone for music.
I have read that Apple discontinued support for the iPod Classic in Apple Music, but I have play counts stuck in the iPod, not to mention 3 devices that are no longer useful. There seems to be 3rd party software that fixes this by circumventing DRM, but I have no desire to do that, and it wouldn't seem in Apple's best interests rather than letting my iPod work.