I think you nailed it. That makes sense. 🙂 An animated GIF is looping, and if we edit a Live Photo and loop it or use the bounce effect, it will be stabilized and will play a lot better as an animated GIF. I turn all my Live Photos either into loops or bounce versions, so I did not notice, that "Export as GIF" is missing for standard Live Photos.. They will wobble less up and down. If possible, I use bounce, because Loop will add transitions, and the transitions will create ghosts - people appearing out of the nowhere and vanishing in thin air.
The restriction to edited Live Photos should be explained in the user guide.