There is no regulation anywhere in the FARs prohibiting a PPL pilot from letting a passenger (which is all your former student was in that scenario) take the controls on a Pt 91 flight. Unless you are doing something well outside of the norm, I can't imagine a FSDO inspector caring even one bit. You school may certainly have a policy against it, but the FAA won't care.
In fact, you don't even have to be a CFI to "teach" in an airplane. A CFI is only required for training toward a certificate, rating, or currency requirements (such as a BFR). Your friendly neighborhood avionics tech can fly with you to "teach" you how to use that cool G1000 you just installed. A lot of specialized "instruction" in unique parts of this industry such as Ag Aviation are done by pilots who may not hold a CFI.
From your other posts, it sounds like there is a lot more to this story. Possibly some widespread misbehavior by students throughout your school. I fully agree that this kind of hazardous attitude needs to be stop punched, even if the students did not break the letter of the law.
Checklist usage is important in small airplanes, both for immediate reasons and for building good habits for the future. Most light training aircraft only need very basic checklists, such as "Controls, Instruments, Gas, Flaps, Trim" or "GUMP". The issue is how many schools have created 50+ page "how to fly" manuals that only distract students from their primary job of flying the airplane. If you can't fit a normal ops checklist in normal sized font on a double sided piece of laminated paper, then you are doing something wrong.