I agree that the probability of our subjects dismantling all the necessary components and moving them to the beach is very low. In reading Betty's notes, and what Betty said during interviews, I'm left with the impression that IF they found Gardner Island, they didn't land on it, they crashed on it. Now all of this is not relevant to proving the theory, it's just fun to think about but picture this:
After finding Gardner she checks the reef flat and wind direction in however many passes it would take to satisfy the pilot. The Electra comes in low and slow, engines muttering like sowing machines, skims the Norwich City and sets it down as soft as she can. As the plane slows during the rollout, the left wheel rolls through the surf as the pilot stands on the right rudder to try to keep the plane strait. As the speed falls off, rudder authority falls away rapidly until the plane goes ballistic and skids toward the broken reef edge and drops the left gear into a hole on the reef edge. The sudden deceleration whips the aircraft to the left, twisting the left gear off the airframe and dropping the left wing into the water at reefs edge. The aircraft is too badly damaged to ever fly again, but not so damaged that the right engine can still be run to charge the battery.
In such a scenario, the cockpit would not be 7 feet above the water, but would be a good deal lower. It has been my experience that equipment of that era was considerably more robust than what you find today. After all, we're not talking about a transistorized cell phone, but equipment with vacuum tubes and wires. I think that the equipment just might work after a dunking, but I also agree that it wouldn't last long if it did run. The above scenario might explain some of the dialog in the alleged radio calls, i.e.: both pilots injured, water rising, "watch that battery" , etc. etc.
Brad