Thanks for your answer.

A missing dependency
libxcb.so wasn't missing. I checked the dependencies using readelf and discovered that libQt5DBus.so is required by xcb. That one was missing, my applications are running now.

The paths can be set using qt.conf which seems a better choice than manipulating the system environment.
I will use qt.conf instead of /etc/environment to tell Qt where to look for plug-ins. But this works only for the plug-in libraries, not for the regular Qt libraries (correct?).
How tells the installer the system where to look for the Qt libraries (I found no qt.conf file)?