The Autho sheet is a legal document in my view, it shows a pilots intentions before the flight, i.e Pilot and pax are identified, flight time is nominated, air exercises are nominated, and following the flight, actual start and stop times are recorded and the flight time is indelibly recorded as P1, P1u/s etc by the nominated pilot. No one else has a right to record those times in their logbook.
What happens if pilot intentions change during flight? (i.e. arrive at airpot and its IMC and someone else takes over for the landing? or feeling knackered so your mate flies). P1 changeover midair is something myself and friends have done numerous times.
You are also assuming everyone has one of these sheets. Our aircraft signout sheet does not stipulate who is doing what, only date, who is taking it, where and acutal hours flown entered when you get back - not before you go. There is no nomination of flight time/hours/exercises/etc and why should there be? What you decide on the ground may not be what you do in the air for whatever reason. Flying clubs may have their own rules.
The logbook is the place to record your hours/responsibilities.