Since when are private individuals hiring an hour in a plane considered to be air carriers or aircraft operators?
The individual is not, but the school, club, person who is renting out the
aircraft is.
Problems arise when PIC confirms all paperwork correct but doesn't know
that the (in date) insurance cover note is invalid as the premium has not
been paid. Although the flight was "illegal" I doubt the PIC broke the law,
and was certainly entitled to record PIC (if not required to under EASA).