The reason gas code does not want you to have a damper for gas fireplace is so that you don’t kill yourself or your family with CO. CO is the product of incomplete combustion, that is the fire does not get enough oxygen, the damper issue is lack of draft, air moves up and out the chimney drawing air to the flame. So that is why. A gas flame that has sufficient oxygen produces no CO. A gas flame that lacks enough oxygen looks yellow, wavy and produces soot.
Looking at your fireplace and the gas log thingy, both appear to be pretty old. Is there a pilot light and pilot safety device such as a thermocouple? That would be there to prevent you from blowing your house up in a gas explosion.
Is your granny 80 years or older? If so, she survived not having all this safety stuff, but she was pretty smart.