Except where his or her presence is not required legally by the ANO.
As are the circumstances here. A 28-day check flight is not required by the ANO, so the instructors presence is not required. Either the instructor logs P1, making it a lesson, or the instructor does not log P1, making him a passenger.
If I checked someone out for an hour or so and then said at the end of the flight. "I am sorry but you handling of the aircraft was not good enough to me to allow you to fly the aircraft solo How would they then log that flight, P1 or Pu/t?
Tricky one. If his license is valid, if his class rating is valid, if his medical is valid and if he is within his 90-day currency, then as far as the law is concerned, he is legal to fly the aircraft as P1. So he can log P1. Doesn't matter if the club requires a checkout above and beyond what the law requires. However, club rules will still prevent him from taking the clubs airplane solo after a failed checkout.
With regards to club checks, I think the situation is actually very simple: A club checkout is not a thing described within the law. It doesn't require an instructor, as far as the law is concerned. There are clubs/groups who allow certain designated experienced PPLs to check out other pilots, for example. So as far as the law is concerned, a club checkout is simply a flight which is undertaken by two people who both happen to have a current and valid PPL (or higher), in a single-crew aircraft. Just as with two PPLs on board, you've got to decide who gets to be PIC for the flight, but only one of the pilots on board can log P1 time. If both of the people on board are hour-builders (one for CPL, the other for ATPL, for instance), I guess it comes down to who can argue the best to resolve the conflict. (Me personally, as the poor PPL who pays for the plane and the instructor, would insist on logging it as P1 and otherwise find another person to check me out, FI or not, within the club rules.)
The only exception is when you make it into a flight lesson. In that case the "extra" person on board has to be a current FI who then logs P1, and the pilot getting a lesson and simultaneously getting a checkout is logging P/UT. At that point in time, this lesson can also count towards CPL license issue requirements, or towards your SEP revalidation requirements. This is the obvious way out for a PPL who is out of 90-day currency. You book a lesson with an FI, do the three landings as sole manipulator, log it as P/UT while the instructor logs P1, and you're good to go flying with passengers as far as the law is concerned. If the club wants to let such a lesson coincide with a club checkout, I can only say that it makes sense.