I may miss something simple here, but I have enabled ftp, created a share, privileged a user to this share (RW) and login successful with this user - just to find an empty root, where I cannot write into.
If you cannot delete a share if because a service is using it. Go back to the reset permission utility and deselect the share you just picked and put none. Of you still cannot delete it is because another service is using it.
is there a view, where you can see, which services are using a share/directory?
Unfortunately no. A common practice is to inspect this file /etc/openmediavault/config.xml shared folders have an assigned uuid. Look where that uuid repeats inside the file, and under which service is being held. After that you can look at the service in the webUI to check that is effectively using the share and remove it from there.