Let me see if I can explain what I mean a little better. In order to work the way it does Tunes (a.k.a "Music" under Catalina) depends upon the app, media files, and a few database files being present. You're not wrong about working with the tracks/media files when adding them to iTunes/Music. As files are added to the music library the database is updated.
The OP's situation is a little different though. The music files, app, and library database were already in place on the dead system. Since the OP had a Time Machine backup, that can be used to move the database files to the new Mac and the updated app is already present on the new machine. It's important to know where the music tracks themselves are stored because two situations can occur:
1.If the music tracks are left stored in the music folder, the tracks are backed up by Time machine and can be restored when the music folder is restored.
2. Since the music files themselcves have been moved elsewhere, in this case an external drive, they are not restored when Time Machine restores the music folder. In that case you have to make sure that iTunes/Music is posted to the drive/folder where the music tracks have been moved.
In the second case Time Machine's default setup does not backup the music tracks themselves. Steps must be taken to back those files up separately.