Having Trouble Restoring Music Library (including Playlists) from Time Machine backup

My music library got messed up during an attempt to clean up my computer and I want to restore my music library from a Time Machine backup. Critically, I need to restore the PLAYLISTS. I do not subscribe to Apple Music. I closed Music, went into Time Machine and restored the "Music Library" file from the "Music" folder, and re-opened Music, but the music library looks the same -- that is, without the playlists that were there at the time of the Time Machine backup. What am I missing? Is there a next step after restoring the "Music Library" file in order to get the missing playlists back? Thanks so much!!