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!