This is somewhat annoying. Found the same problem even when trying to feed miners by inserter from a belted output. In the code, there is a list of entity types that are allowed to be fed by inserters, which is lacking mining drills. I assume this is to prevent electric mining drills from being fed by inserters, and then evaporating the item as the drill is removed (inserters that "deactivate" destroy their held items). I don't think this should be a problem though. Editing it on my end now~
EDIT: As far as I can tell, it worked. Haven't tried burner circles, but inserting into burner drills does work. Add this line somewhere among the other usefull_target_types in control.lua:
usefull_target_type["mining-drill"] = true