From my instructors background: a rental checkout is either a demonstration of proficiency ( type flown previously) or training to proficiency ( type not flown previously).
Or combination of both at the discretion of the instructor really.
In any case it depends on the certification held by the pilot undergoing the checkout that determines who is PIC.
Unrelated point of interest.
Being an “airline pilot with a bazillion hours” doesn’t mean they’re safe to fly a “little” airplane.
Those bazillion hours in a transport category airplane do not carry over well to a single system single engine aircraft with a fraction of the performance they are used to.
Flew with a retired airline pilot guy once who told me that he lost 3 former colleagues in a year that went back to flying little airplanes after retirement.