I have had this issue with iTunes in High Sierra as well hoping to get some help now that the issue persists on Catalina and the new MacOS Music App.
If I hear a song I like in a playlist I use the "…" select add to library, it adds the song but then stops playback of the song and exits the playlist. I have Apple Music, Running Latest MacOS (10.15) and Latest Music App (Version 22.214.171.1246)
- Open Apple Music playlist
- Use … to add song to library
- Song will stop and playlist will exit