I cannot get the action "export cover to file" to work properly

You have to tick the option not to export any duplicates.
Still, if the selected files of the same folder that have varying embedded pictures then all of them get exported but they do not overwrite each other but get get renamed.

Hard to say what is happening there without further information.
A good guideline can be found here:

A help could be to apply a filter to really treat only files that have an embedded cover:
%_covers% PRESENT

or the other way round - see the files without embedded pictures:
%_covers% MISSING