Thanks for all of the suggestions.
I finally bit the bullet and brought the trailer to the fabrication shop. As it happened, he had his own, larger dump trailer parked there. We decided to mostly copy the design on that tailgate. It's similar to the suggestions and photos some of you showed in this thread.
He made hinges and mounted them on rear edge of the top of each side.
Built a frame matching the profile of the rear of the trailer. The top of the frame is attached to the hinges. The bottom is held in place by self-locking toggle latches (not sure if I've got the name of those correct?). Replaced the original side hinges with some redesigned ones to improve clearance while in use. Those new hinges work much more smoothly than the original and are greaseable. Those side hinges and the original latch plate (in the bottom right corner) were moved from the rear of the trailer to the newly made frame. When used as the original door-type tailgate, the frame stays latched to the rear of the trailer.
When I want to tailgate out some gravel, the door is latched shut to the frame (using the original spring-loaded latch pin on the lower right, with the catch now mounted on the frame). The toggle latches are released and the check chains adjusted for the amount of opening needed. The gate opens by gravity as the trailer is dumped.
I've only used it a few times since the modification has been done. It works very well on driveway gravel. I have not tired it yet on the 1.5" stone I use as a base or to fix up some of the low/wet spots on some of my woods trails, but I'm sure it will work fine as soon as I figure out the proper check chain setting.
Learned the hard way that it does not work well on wood chips (at least not when they are green/damp): they slid together as a block and just jam up the opening, rather than dumping out in a smooth stream. Fortunately, wood chips are easy to rake smooth.