"Load lyrics beside music file" not working

Prerequisites

LyricsX → Preferences → General
- Preferred music player: iTunes (Note: called Apple Music for a while now)
- Lyrics saving path: LyricsX (Default) which lives at ~/Music/LyricsX/
- Load lyrics beside music file: ON
- Keep Music Media folder organised: ON
Apple Music → Files → Music Media folder location
- I do not use the standard location!
- But I use a custom location: /Volumes/my-music/Shared/Music/Media/Music which I refer to as `MusicRoot'
- Note: Maybe that custom media library location is the cause or at least related to the bug?
At MusicRoot I have the following folder/audio-file structure (managed by Apple Music) with .LYR files added by another app than LyricsX and some self-created lyrics files.
- Artist A
- Album X
  - 01 Song One.mp3
  - 01 Song One.lyr
  - 02 Song Two.mp3
  - 02 Song Two.lyr
  - ...
- Artist B
  - Album Y
  - 01 Song Uno.mp3
  - 01 Song Uno.lyr
  - 02 Song Duo.mp3
  - 02 Song Duo.lyr
  - ...

Start to play a song which has a music file and a lyrics file besides it (aka sidecar file) in the filesystem.
- Expected: The corresponding Lyrics show in the Karaoke Lyrics overlay which I have activated.
- Actual: No lyrics are shown.

I open the Lyrics Window, which says: "No Lyrics. Drag & Drop to import LRC file"
- I drag and drop the LRC file manually from Finder and from now on the synchronized lyrics show just fine.
- After playback of the current song is stopped, a corresponding .LYRX file gets copied to ~/Music/LyricsX/.
- It's a duplicate. Almost. Only difference I noted: The original LYR has centiseconds (.000) and the created LYRX has milliseconds (.000).
A also did the Reproduction again with a new song with the Internet connection off, so rule out a conflict of the automatic internet lyrics lookup vs. the local sidecar file lookup.
- Made no difference. Same problem

Your app has some hardcoded filepaths which are troubled with a custom media library location.
File access permission issues? If so, what would I need to grant and where?