You can delete the avatar via:
I haven't tested it to see if it goes to the 2nd uploaded avatar if it exists or the system default...looks like system default from the description, but if it goes to the 2nd one you can just make the same REST call again and it'll delete the second one.
I think in 3.11 of the API the avatar would not update. Not sure if the person service even will remove it in 3.11.
I recall it being an issue so there is a good chance you cannot do it in that version.
Question here as well that got missed: Jive API : Deleting Avatar returns Status code 405 "Method Not Allowed"