the much-photographed hulk on the shore of Gardner Island was the British-registered freighter SS Norwich City, which ran aground in the early 1930's I believe. Now an American would naturally pronounce 'Norwich' as 'Nor Witch rather than the correct British pronunciation of Norridge. With a dodgy radio link, this may well have come over as 'New York City'. I suspect that in 1937 the ship's name was still legible. There are I believe bits of the ship still visible on Gardner.
If true, this transmission would I think go some way to confirming that the Electra came down on, or partly on, Gardner Island. Which, rather than being a total desert island, had been inhabited for periods of time prior to Amelia's disappearance, and was again afterwards. I have a feeling that I read one time that there is a fresh-water spring somewhere on the island.
I don't think the Japanese PoW theory holds water, in spite of fairly recent claims for it, and I honestly don't think the disappearance was faked. If it was faked, for what purpose?
That Japanese story actually dates back fairly early after the tragedy occurred. In 1943 a film starring Rosalind Russell and Fred MacMurray called "Flight For Freedom" appeared in Hollywood, and was a really badly disguised story of Earhart, and suggested she was on a spying mission of the Japanese military installations on the last flight, but when she finds they are going to force her down and capture her she purposely (ironically) "kamikaze" crashes to successfully complete her mission.
The evidence for that solution is as bad as the evidence for the New Jersey disappearance (why indeed disappear to restart her life? - she and George Putnam were not really lovers, but formed an affectionate partnership based on her writing career, and he really was deeply involved in the search attempts).
The Gardiner Island theory is the best one to work on for any potential proof. The Pacific Ocean crash would be the most difficult (where to search for the plane?)!