OK, so to clarify:
Family Sharing is a free service that allows you to share your purchases from the iTunes & App Stores with other members of your Family. It also provides a common method for paying for all future purchases via the Organizer of the Family, in addition to providing a common place to share a calendar or photos.
There are two additional services under Family Sharing:
- iCloud Storage Family subscription
- Apple Music Family Plan subscription
You can use Family Sharing without having either a shared iCloud Storage plan, or a shared Apple Music Plan.
So, that is why I asked for the clarification.
Since you are using Apple Music, none of the items related to Apple Music are backed up via an iCloud backup. That's because Apple Music is a streaming Music service, and by its nature, it is an iCloud-based service. Therefore, nothing that is used on that service is copied to an iCloud backup of the iOS device.
The Playlists live in the iCloud Music Library, which is a single library that is updated dynamically when you make changes, purchases, deletions, and other modifications to your Apple Music. I'm fairly certain that the playlist is gone, but here are some Apple Music articles that may help:
Create and share playlists - Apple Support
Access your music collection on all of your devices with Apple Music - Apple Support
Get an Apple Music Family subscription - Apple Support
I'm afraid I couldn't find anything about retrieving deleted playlists, but again, if it was in the Library, then it was probably deleted from both places at the same time.
Best of luck,
GB