Use Sync Library to stream your music library on any device that's signed in to the Apple Music app with the Apple ID that you use with your Apple Music subscription. Learn what you need to use Sync Library and how to turn it on.
support.apple.com
I’m not aware if a utility such as 3uTools does media management, but you can look into ti.
The sync button only appears in music settings if you subscribe which I did but why isn't Apple got sync option without you having to subscribe ? The only other way is to sync if you have a computer .