Upfront I will admit to being a newbie late to the discussion and only slightly read in to the overwhelming amount of info on this site but as an aviator would like to post some thoughts for consideration in this debate. Having frequently been involved in extended radio checks I know that after a certain amount of "testing, 1,2,3..." etc. you often find yourself resorting to words that are nonsensical. Without regard to the validity of Betty's notebook I would not be surprised to hear unusual statements on an extended transmission like "New York, New York" if someone were trying to insure a steady, long transmission to faciltate direction finding. After several days of not receiving responses at some point you would suspect your receiver and hope that at least you were transmitting and someone could get a bearing on you to come help. You would transmit info you thought was important but to extend the transmission add whatever filler came to mind.
As to their physical condition under the hypothesis I would wonder how either of them could be seriously injured if the aircraft was not sufficiently damaged as to allow an engine to be run and the radio to work? I would instead suspect some other injury resulting from the process of exploring/searching for shelter/water. Some of these injuries can result in less than coherent behavior such as drinking contaminated or salt water, cutting one's feet on coral, eating something bad. Sunburn, sunstroke, and dehydration all also produce unusual behavior. So silly babbling would not be unusual after a while.
Apologize for going off this thread but in line with that train of thought, as an aviator I would have circled the island looking for the best place to land. In that process I would have noted all the features of the island to focus my immediate survival efforts. If, as stated, the dark pond at the SE end of the island looks like a fresh water pond I would have gone there. Seems others assumed till they found out otherwise that it might be a fresh water pond. I would have had to taste the water to find out it wasn't fresh and who knows what that might have done to me? I would have also noted all open/cleared areas as having the potential for inhabitants. We know the island was not inhabited, they would not have. They see a wrecked ship and might expect someone to still be there. If you saw a clearing such as the 7 sight that might be the next place you look after finding no one near the wreck.
Finally, as an aviator I do not doubt in the least your hypothesis about what they did upon arrival at LOP. That decision would have been made before hand, if not by AE at least by FN, who knew the only direction to go. After a brief run to verify to the north you would go 337 toward the greater preponderance of islands close by. Having been low on fuel and "sucking up the seat cushion" as we say I also know you would land at the first place you found suitable once you are flying on reserve. I know from experience flying over water that islands can sometimes be seen at 100 miles that at other times cannot be seen till you are right on top of them. Humidity, sun angle, and cloud cover all play a part and sometimes low cumulus is worst as the shadows can hide a sizable island from view. They would have arrived at Gardner with sufficient fuel to conduct a survey similar to your helicopter ride. Prudence would dictate a low approach to determine the texture of the landing surface. What looks smooth at 100+ feet can turn out to be a washboard on landing.
I applaud your work to date, am fascinated at the analysis and wealth of data on your site which I have only scratched the surface, and apologize if I have added nothing to the debate that may be found elsewhere that I have not read yet.
JB