I'm afraid the main issue here was as dump as it was simple.

I just did not run the rsync ... command as root:

So rsync set the owner of all directories an files to UID=1000 (pi) instead of UID=0 (root), so there was of course trouble with the permissions.

The answer of @Milliways was still very helpful to understand some stuff. Thanks!

I'm afraid the main issue here was as dumb as it was simple.

I just did not run the rsync ... command as root:

So rsync set the owner of all directories an files to UID=1000 (pi) instead of UID=0 (root), so there was of course trouble with the permissions.

The answer of @Milliways was still very helpful to understand some stuff. Thanks!