Apple Music Sync Library option is greyed out

Are you a subscriber to Apple Music or to iTunes Match? My understanding is that "Sync Library" (a.k.a. "iCloud Music Library") is only available if you are a subscriber to one of those services.

Use Sync Library with your Apple Music subscription – Apple Support

"Sync Library requires a subscription to Apple Music. You can also use Sync Library with an iTunes Match subscription."

If you are not a subscriber, or do not want to use "Sync Library", there is the traditional "manual" synchronization model first introduced with the iPod.

Sync your iPhone, iPad, or iPod using your computer – Apple Support