Iāve spent some more hours by trying around, especially as I have - or better had - one working NC instance. Tracking the differences Iāve found that the only one was that the working NC instance was sitting behind an older HAPROXY version than the other ones.
Iāve updated that HAPROXY and it stopped working here as well
So Collabora introduced an issue with 6.4.10.x in case the Nexcloud instance is handled by HAPROXY >=2.x.
Iāve opened an issue: https://github.com/CollaboraOnline/online/issues/3199