An update.
The CD I used for testing seems to be problematic. No scratches and no issues in other players but with this particular drive/laser it causes the strange behaviour I have reported.
Shifting to other CDs (can't believe I did not try that before), I found that it works fine - except that it is now and then stuttering as if the CD is scratched when it is clearly not. Like it is having trouble tracking or something. I believe Dillen suggested in an old thread that this can be caused by a not perfectly smooth movement of the laser carriage. It might be that the gears and the rail need some extremely thorough cleaning with an even greater attention to detail than I had when I did it before. Or maybe there are other reasons behind the skipping.
In any case, I will not troubleshoot this further until I may need to use that drive or laser in another future project. I found a CDM4 based drive and PCB on eBay that swapped right into the Ouverture without any further modifications. The mechanism is a bit slower to move between tracks on the CD, sure, but it just works and I have the same type in my Beocenter 9500, which I like the sound of very much.
So, although I have not found what caused the initial problem, I consider the case closed, for now.
Thanks a lot for the feedback provided. It warms my heart when you fellow forum dwellers selflessly lend me a bit of your time. I will try to do the same in return. :-)