This site last updated Article 265.20 on 17 February, 2012. I can assure you that while teaching Hunter's Safety back in the late '70s through the early 80's, at the NYS DEC's Stony Kill Farm, familiarity was a routine part of the class. No live-fire, as there was no shooting range on site. However, at the same time (and through to the early 90's), live-fire was a mandatory part of our classes for Handgun Safety (then a requirement for all CCW applicants) being taught at the Dutchess County Pistol Assn. range.
Your restrictions may be unique to the County you're teaching in, as certain regulations/restrictions may be imposed by the issuing authority. The only local shop we have nearby with the badge/permit requirement before handling is Gander. That seems to be a company policy for them, but it could very well be an interpretation of some regulation pertaining to the FFL holder, rather than the consumer.
As to the above Gent's comment about C&C permits for each county, I'm confused as to his meaning. A permit issued anywhere in the state is valid throughout the entire state, excluding the five boroughs of NYC. They, of course, only recognize their own permits, which are also good throughout the remainder of New York.