I tried to come up with a witty analogy, and they all have holes.
- imagine the hummingbird/drone/helicopter is in a stream and wants to stay stationary
– imagine the h/d/h is in space, but is caught in the gravity of something other than Earth, that just happens to be accelerating it at 9.8 m/s/s
– imagine the h/d/h is twice as heavy, etc….
So I resorted to google, “physics of hovering” gives some good links
Amount of energy required to hover is good but assumes the air is giving a gliding bird the energy instead, as I understand it. That link has a link to why does holding something up cost energy while no work is being done.
From the abstract, The physics of flight: III. Hovering sounds like a paper created very similar to your own search. But I didn’t access it ;)
Regarding the work done, I think it depends on how big you make your system. No work is being done on the overall system, but in reality the lack of motion is because work is being done in smaller pieces of the system. Rotors are turning and wings are flapping to create the lift needed to offset gravity.
How much energy is needed really gets into efficiency, but it sounded like you were looking for a generalized answer.