Ohhhhh.....
Yes, connecting to chronyd over the network worked. Next thing, I tried connecting to itself using -h 127.0.0.1, which worked fine. Connecting to itself using its hostname, however, didn't work, which finally pushed me in the right direction:
in /etc/hosts, there was an entry
which, as we all know, is not the default address for localhost. Setting it to 127.0.0.1 solved the issue. I guess chronyc defaults to "localhost", if no hostname is given. Then, I guess, in some other part of the software only 127.0.0.1 is allowed per default; not other loopback IP addresses.
In the end, this just explains the cronyc/cronyd communication difficulties, right? cronyd should have still run without issues...
Anyway, thank you for your help :)
Benjamin