Actually I’m not, as they’re legal for off road use eg on the disused airfields at the Newark Runway Monday Antique Fair or Lincoln Antique Fair.
You're making a rookie error.
It is not ownership of land that determines whether or not it is "private" for the purposes of the Road Traffic Act. It is who has access to it at the material time that determines its status as public or private.
A pub car park may be privately owned but still open to the public and thus part of then highway during opening hours. Ditto a supermarket car park.
This concept extends to antique fairs - if its open to the public then irrespective of the ownership status of the land it is a "public place" for the purposes of the RTA. You run someone over there you'll still get knocked off for all the document offences (licence, MOT, insurance) and quite likely for careless or dangerous driving. It only becomes "private" in this context when the public have all gone home and the access is closed.
If the device is "readily" adapted or switchable then it is illegal, regardless of what mode it might be in it the time. As Regular Cyclists advises, removing the throttle and not having it on your person means it isn't readily adaptable and is nominally legal.
If you want an electric moped or speed pedelec then go any buy one and be sure to get the licence, insurance to go along with it. We have a hard enough time as a user group already without this sort of thing.
Love, Drago (retired Roads Policing plod)