I would like to add all of the emptys as individual entries in the asset browser.
At the moment, Asset Browser still lacks the functionality to use object groups(Collections) as assets. As this functionality is essential it would be reasonable to expect it added in the future and it is in the goals as we can see here: https://developer.blender.org/T94031
So at the moment full functionality needed for what you want is simply not there. You could use Python to make collections out objects parented to the empties and then make instances of those collections that can be made into assets, however it seems that you would need to somehow generate thumbnails as well.