Let's review this in the whole sense: PPL shows up to rent a plane. PPL is licensed to fly the aircraft, the licensing authority has no further interest in the PPL demonstrating their skill - they have the PPL. The PPL is confident about their skill, and prepared to take the plane solo - they do not request a check flight. If they did, they would be requesting training, and thus flying PuT with an instructor - but that's not this from the PPL's perspective.
But, the aircraft owner would like the PPL to demonstrate their recency under their commercial 28 day rule. They require an instructor to ride through a check flight with PPL. The PPL has no intention of being interceded by the instructor during that flight.
The PPL has no intention of sharing the flying duties with the instructor, as they came to the dispatch counter prepared to sign for sole responsibility of the aircraft, and to fly it unassisted. Therefore, they did not come to the flight seeking "help" with flying the plane, but they are willing to follow the rules, and allow an passenger instructor to evaluate their skills for currency.
So;
yet still retain the RIGHT to intercede in the control of the aircraft. To follow your logic let us say, for example, you intercede incorrectly and there is an accident. Do you think that the "P1" and his lawyers would be happy for him to shoulder all of the blame and for you to walk away scott free?!!!!
Yes. The PPL came prepared to fly the plane solo, not to request training. So there would be no "right" for the other person to intercede in flying the plane. If they do, they are not interceding, they are interfering. Acting in the employ of the aircraft owner, they should really do know better. If they interfere, and make things worse, they will have to take the responsibility for that - the PPL renter did not accept the responsibility for the aircraft with the intent of assessing the ability of the passenger instructor to interfere, without creating a further hazard. They're just prepared to manage their own skills.
There have certainly been cases where an instructor wanted to "show" a PPL something during a check flight - I've been to a few of the accidents. The maneuver was beyond the required skill set, and beyond the PPL's skills. The instructor took control, and responsibility.
The role of the check pilot during the check flight is to
assess the PPL, not to train, or "help" to fly the plane. If they have any
need to touch
any control, the check flight was a fail, and it's time to go home to reassess.
So if the instructor is along to assess, and not to fly, they're not flying if they're doing their job right. If they are not flying an SEP, then someone else is, so that someone else - the PPL - must be the PIC!
No disrespect taken, nor intended at instructors - who I know are just doing the job they were assigned. I opine that the job may have been poorly defined form the outset..... The renter PPL is PIC during a check flight, because they are flying - that's the purpose of the flight!