I often latch the hitch head in the coupler when camping. After unhooking I hold the ball up in the coupler and have my wife lock the coupler latch. It keeps the hitch head out of the way and the kids don't get grease all over themselves when they inevitably brush up against it.
I wouldn't park the trailer in storage with the hitch head attached to the trailer, as you said, too easy to steal. I keep a 36x36' sheet of 1/4 plywood that I put on the floor of the trailer when it's storred, the hitch head, bars and gas grill sit on the plywood to keep the floor clean. It's not far from the door of the trailer to the hitch on the truck so I don't mind carrying the hitch head that far, and since I never set it on the ground I don't have to lift it, just carry it.
1977 Safari Land Yacht
2005 Toyota Tundra SR5
2010 Ford Flex Ecoboost