Hi John. It's common practice to tie one side the DC source to chassis ground. Use of a green wire is arbitrary. They can do it at the battery
or the converter. Back in the good old days when electrical
equipment was housed in metal chassis, tying all that metal together was a safety issue. With plastic cases that's not an issue any more.
The best reason I can give you is that it reduces the possibility of parasitic effects. An electrical
circuit can behave as designed on the test bench, but installed, can behave quite differently due to long wires, proximity to other electronics, and a multitude of other things. For example, power supplies can turn into radio transmitters. That sort of thing. Good grounding reduces that kind of unwanted behavior. Will your DC appliances work without it. I suspect so. But if all of a sudden you LEDs dim or you get a fuzzy picture on you TV, poor grounding could be the cause. Raz