It's awfully difficult to know what FN would have done - in fact, impossible in my view. Gary has cited from reliable text material of the day, I believe, very often in his suppositions of these things - and I guess the box search is a method that was known.
You make good points though in that a greater set of possibilities existed, and in full context it should not be lost on the observer that the Phoenix group did offer some fair chance of alternate land fall. I don't know what FN knew, but I suspect the fact of the Phoenix group was not lost on that navigator; similarly, I find it difficult to believe that the thought of that as a possible alternative would have been overlooked. The navy didn't think so in 1937 - and had bothered to know what FN's contemporaries believed about the navigator's habits. Friedell applied the logic of a landplane navigator instinctively looking for land, as opposed to the sea navigator avoiding same, in time of need.
How well charted the Phoenix islands were I'm not sure. I believe Gardner (now Niku) was not accurately portrayed on the chart of the day. Nonetheless, there it lay - a 'Coronado Strand' in appearance per Lambrecht, who overflew it. Big, bright lagoon, 90 foot trees and all. True, those islands are scattered - but they exist, and there's a better chance of landfall to SSE on LOP than to NNW, and a better chance of seeing something like Niku than Howland if in any reasonable proximity.
'Pancake' isn't a bad way to see Howland - if it can be seen at all from any reasonable distance. None can say what 'Fred would have done' with any accuracy in my opinion, but one thing does stand out to me: given all that led up to the "on the line 157-337" call and what the books tell us of LOP navigation, and by what can be understood from Friedell's logic as to the landplane navigator, flying the line NW, thence not having found land, SE with hope - and with the Phoenix group further down as back-up if needed, a box search sounds like a loser. Just MHO, of course.
I'm not the navigator Gary is, but that isn't the point. The real point is that FN can't be channeled and the box search is only one opinion, no matter how well Gary knows the subject matter. How well it is supported in terms of what FN would likely have done must be up to the observer. Not sure it can carry more than that unless the bird is found and supports the idea.