You seem to want to continue with the AME analogy, despite it having utterly no relevance to a discussion about a PPL training flight.
His function is to ascertain the applicant’s knowledge and skills
So if you ascertain that the person you are flying with is utterly incapable. You just leave it at that? Aye right. Only an utter moron would think that an acceptable situation.
and interject if necessary to improve on these.
Where the aim is achieved the FI will sign the applicants logbook
So if the aim of the flight is to ascertain the level of knowledge and skill and improve them, but you find these to be woeful and don't show improvement, then the aim of the flight hasn't been achieved.
Therefore, no sign off.