Copying/moving music library to a different Mac loses all playlists
I am trying to move my music library to a freshly installed MacMini running MacOS Ventura 13.2. The music is stored on an external USB-drive which I connected to the new Mac. The Music app finds all the albums and titles. All my playlist also appear, but they are empty. I have already tried to copy the "Music Library.musiclibrary" from the old to the new Mac, to no avail. Also exporting the database to .xml and importing did not solve my issue. Any hint how to restore the content of the playlists?
Some details: my music library contains some 10'000 titles, all from my own sources, nothing from Apple Music. The playlists are organized in playlist folders. The older Mac is running the latest version of Monterey. The playlist still works on the old Mac, I also have TimeMachine backups available.
Many thanks for any advice
Mac mini 2018 or later