As a long time chief instructor for a major state agency and the state academy I have to say that anyone who advises using fired cases for snap caps is someone that I wouldn't utilize for any instruction. First, they are difficult to identify from live rounds by any and all students. Second the act of dry firing will cause the case to get bumped farther and farther into the pistol chamber creating problems of several varieties or causing pieces of the primer or case to cause this type of problem. Third, all competent and professional instructors should have snap caps either home made and painted red or professionally made available for those drills that call for them during the instruction process. Fourth, when doing dry fire drills, there should be nothing that even looks like live ammunition in the classroom or in the area while dry firing is being accomplished so that everyone can see that the weapons are in fact clear.
While I understand the need to dry fire during training and recommend it, I have to say that most modern handguns don't strike the firing pin with sufficient force to damage the firing pin. So the mere act of dry firing will probably not damage the pistol unless you are doing many hundreds of dry fires or more.