The requirement for a licensed aerodrome relates only to training and testing for a licence or aircraft rating. Most FTOs are enganged in such training and so are normally located on licensed sites. However, there is no reason that an FTO should not operate from an unlicensed site so long as it restricted its training from that site to, for example, IR and FI courses.
ISTR that there was a helicopter FTO located at an unlicensed site adjacent to Manston some years ago.