I am having this exact same issue. I have tried troubleshooting steps like toggling the iCloud Music Library, disabling Add Playlist Songs, toggling Cellular Data, verified my Apple ID is logged in... nothing is working. I am still unable to add songs to playlists or delete songs from playlists.
When I add a song to a playlist, the icon with the check mark populates saying "Song added to playlist" but it fails to populate.
Additionally, when I delete a song from a playlist, it removes it from the list and the "loading circle" spins and spins but never stops spinning. If I back out of the playlist and go back in, the song is still there.
Please help! This is very frustrating.