Good thing I didn't get around to taking [name] to the range, because what nephew LEO told me was flat-out WRONG. In my state, it's a felony offense to carry a handgun, even unloaded, outside the house (even on my own property) without a CHL. The only exceptions are transporting (not "carrying") from the shop where bought or to a shop to sell; to a gunsmith for repair; or to or from MY (not my employer's) place of business.
I will not say your state does not allow it, not knowing your state, but very few states, I know of, prohibit you transporting a lawfully owned gun to a range. Usually it falls under the same exceptions you list above about shops and gunsmiths, hunting. In many states it would have to be cased and locked in a trunk or similar arrangement and would have to be cased when carrying to the car.