Music app only shows an outdated music library after I've retrieved it from backup
Thanks to Apple Support I was able to solve it. If you haven't saved your music library in iCloud, you need to retrieve the iTunes folder from your backup (or generally the one where all your songs are saved) and another folder in the same directory, Music, which should contain a single file, Music Library, that's up to date.
Close the Music app (if it was open) and reopen it while pressing the option key (alt). Navigate to the above-mentioned file Music Library and you've got your latest library back.