I was presuming that they went around McKean and Gardner more than once, since the first sentence talks about "one circle" and then "further inspection."I see. Thank you.

It would appear that you're extrapolating from the comments about McKean ("perfectly flat... with no vegetation whatsoever") and assuming the aviators took the same approach and actions at Gardner - a
much larger and more complex location. If your orders are to "search", you're going to have to do more at Gardner than you would at McKean in order to comply with those orders.
The "repeated circling and zooming" took place over the "signs of recent habitation."Another assumption that isn't directly indicated by the report, but a logical one that is a definite possibility. But an equally possible assumption is that, having found signs of "recent habitation", they not only zoomed and circled that site, but took a closer look at the rest of the atoll too, just in case the castaways were elsewhere; looking for food and water, gathering firewood, etc.
Lt. Lambrecht's comments about the only other uninhabited island that showed "signs of recent habitation" (Sydney Island) indicated they made "several circles of the island" and "repeated zooms" without eliciting any response. As long as we're extrapolating, that would tend to suggest that they took similar steps at the even larger Gardner Island, as opposed to the more cursory search of the much smaller and relatively featureless McKean Island.
To my eye, the actual count of visits to the island by aircraft vs. the number of ships engaged in the search discredits your certitude.If you count the number of aircraft involved in the search - the three flown off the Colorado, the PBY (even though it never reached the area, it was dispatched as part of the search effort), and all the aircraft flown off the Lexington, the amount of aircraft involved actually greatly exceeded the number of ships involved.
Facts:
Aircraft visited and searched Gardner Island.
Ships did not.
The vast majority of the areas that were covered in the search
were searched by air, not from the deck of a ship.
AE did things that many other aviators would not have done. Would you consistently give the wrong frequencies of your equipment to people who had to use those frequencies to help you land safely at Howland Island? Would you ask for a transmission on 7500 kcs for equipment that was limited to lower frequencies? "Direction finder on plane covers range of about 200 to 1400 kHz." Would you transmit on 3105 kcs if the Coast Guard told you, "Itasca direction finder range 550 to 270 kHz"? Would you transmit too briefly for a direction finder to get a bearing on you?No sir, I would not have done those things. However, I am an audio engineer with decades of communications experience and training; unlike most people, I deal with different frequencies on a daily basis. But I take your point. AE made some significant mistakes. The failure of AE to avail herself of proper training on the operation of the Bendix RDF, or to perform test flights in order to test that equipment and practice with it, along with the decision to leave the trailing antenna behind were significant factors in their ultimate demise IMHO.
Do I think they would believe in an aerial search strongly enough to not search the island or relocate their camp if they found a better place for it? No, I do not think that.No sir, nor do I. "Believing" in the possibility of an aerial search does not automatically rule out relocating (or searching for water, etc.); the two are not mutually exclusive. However, it would probably influence your actions. In other words, the
assumption is that you would conduct that relocation and / or search for water / food with one eye scanning the horizon and sky, and that you would make an effort to leave yourself a ready route to a location where you could be easily spotted or attempt to signal a passing ship or aircraft.
In light of some of the illogical and unwise decisions AE made, it is certainly possible that she put herself into a position where, in spite of the repeated efforts of the naval aviators to locate her and make their presence known, she was unable to reach a clearing or light a signal / smoke fire in time to make her presence known to them. It is also possible, as I mentioned in my earlier posts, that she and FN were incapacitated and incapable of making their presence known. Or that they were never on Niku to begin with - although the preponderance of the evidence currently available would tend to argue against that last possibility.
I think that one visit from one flight of aircraft that failed to spot them (a common occurrence, even in our vastly advanced SAR experience!) would probably not cause them to stay put near the leeward reef.Yes, it's not unheard of for modern SAR efforts to miss people or downed aircraft, as the recent Steve Fossett search clearly demonstrates. But if you're alive, and want to be found, and are taking steps to make yourself more visible (signal panels on the beach, signal fires readied in the event a passing ship is spotted, etc.), the chances of you being spotted are greatly increased.
Again, I am not suggesting that they would remain near the leeward reef; only that is is reasonable to assume they would have taken some efforts to make their presence known in the event a search party should arrive in the vicinity, and that they most likely would have been watching and listening for such search activity.
The evidence would seem to suggest that they eventually relocated to the southeastern part of the atoll; it does not, as far as I know, indicate exactly when that relocation took place. If we accept the northwestern reef landing site theory as accurate, then the only thing we can reasonably surmise is that at least one of them remained at that location for as long as the radio on the Electra was being used to send out distress calls; probably until at least 5 July 1937. If they began exploring and searching the island on 6 July, that leaves only two to three days to reach the far end of the island. Considering the density of the native vegetation, that would tend to argue against an extensive search of the island occurring along the way, with their arrival at the Seven Site location prior to the aerial search on 9 July.
If they were at the Seven Site when the aircraft appeared, they could have reached the beach or lagoon fairly quickly unless they were disabled, thus greatly increasing their chances of being spotted. In my opinion, that would tend to argue against them being at that location by 9 July.