Hey Gary...Adam here...want to tell you I appreciate your putting forth your ideas to critique, and I always respect when people with expertise bring their ideas to bear.
I try to look at things from the logistical/practical ground view, so to speak, and I want to tell you why I, as an admitted aviation know-nothing, find the search pattern theory unpersuasive.
Here's the thing: you have a navigator who's been up for 24 hours straight, there's limited fuel and...here's the kicker: the noise of the engine makes complex direct two-way communication well-nigh impossible. So, yes, in THEORY, one could execute a search pattern. But success or failure would depend on making -- and correct me if I'm wrong -- a series of course corrections that would require careful keeping track of where one thought one was, and how one progressed in relation to it.
I would submit that, under the circumstances, this just would not have been practical in a real world way. Yes, of course FN was a competent navigator, and he'd been passing AE written course corrections. But now everyone's tired, stressed, time is running out, and if FN wanted to embark on this course of action, first he would have to explain/persuade it to AE, and then execute the course corrections calmly and carefully. To me, in that kind of a situation, there's just too much margin of error. What if AE doesn't understand what she's doing? What if out of stress or fatigue, someone screws up a course heading, or just fails to keep proper track of the last turn? Without real time two way communication, there's no way to really work these details out in a crunch situation. And my understand is this did not exist. They wrote each other on a blackboard. So any communications have to be simple and direct.
I anticipate that your response will probably be of course they could have done it...and I agree with you, it is possible. But it's one of those things where I just don't think under the circumstances, sitting in FN's seat, it would have seemed like a good move. Far simpler and more goofproof to tell AE to fly the LOP to the best of one's ability, and use the southern islands as a backstop, and it may have seemed just as likely to get them to land as any other option. I'm not suggesting these people were incompetent...but tired and stressed competent people make stupid mistakes based on bad communication all the time, and that's without having the roar of an engine drowning out any details. And competent people know this, and compensate for it. I don't want to say what FN "would" have done, but I am sure that, having been with AE and in that plane for a long time, there is no way he could fail take all this into account.
That's my opinion, and I don't vouchsafe it as being anything else. I'm neither pilot nor navigator, though I am a student of how things go wrong when people fail to communicate. But to me, from my admittedly surface understanding of what you're suggesting, it just doesn't take into proper account conditions in the cockpit as they existed at the time, and the practical limits of what people can do with limited time in an emergency situation without good two-way real time communication to work out misunderstandings or make a contingency plan. To me, whatever they did, given the limits in time, extreme likelihood of error owing to fatigue, stress and fear, and the inability to really communicate, it would have had to be kept simple.