There lies another problem, Genghis. A large number of club hacks either don't have a POH or it is locked in the CFI's office. Shocking behaviour but reality. I have worked at a number of clubs and it is worrying to find that the vast majority of students don't even know what a POH/AFM is.
Any instructor that doesn't insist his or her student own a copy of the AFM/POH, isn't worth their weight in wet burlap.
How can an instructor possibly let a student go who can't work the performance charts in the airplane, and who doesn't know the systems and procedures in the airplane?
Very, very unprofessional. Forget whether the government or the flying club requires the AFM to be in the airplane. An instructor who doesn't insist on the student knowing the flight manual intimately, isn't worth much.