You clearly do not understand the meaning of the word 'supervision', SteveR. Undergoing a 'check ride' prior to the hire of an aeroplane means that the person conducting the check ride is required to be there in order to assess that your performance is satisfactory for whatever requirements the organisation offering the ac for hire has deemed appropriate. Otherwise there would be no need for a 'check ride' in the first place.
If you are under any FI supervision, you are not the Commander and may only log your flying time for that flight as P1U/S if the FI also countersigns your personal logbook including his/her licence number.
What exactly is the problem with that? Apart from delusions of grandeur on behalf of the pilot under supervision.