Unfortunately no change. Still the status as originally described and shown in the screenshots above.
I actually did a factory reset (10 seconds button hold on the Fairbuds case, then close the box for 5 seconds) before renewed attempts. I deleted the app, then reinstalled it (still 2.0) – from Murena’s App Lounge on the /e/OS FP3+, then from Aurora Store on the Fairphone OS FP4 (I do not have a Google Play Store account). I then tried the same (for the very first time) on the Fairphone OS FP5 with Fairbuds installed through Aurora Store. Same result. I also chose the most “permissive” settings for each attempt now (most generous permissions possible).
Just to mention it: I can listen to music with the Fairbuds, although I tend to think that it’s the pre-firmware sound (could be shinier in the highest tones).