Sorry, but I'm not sure how what you are saying to the subject of this thread: "..So many of these [thyroid cancer] diagnoses are wrong". I would like to have more information about this. If thyroid cancers are often misdiagnosed then the word needs to get out!!
To your point, I am not sure if even the Iodine doctors (Flechas, ...) would implicate Iodine deficiency as a leading cause of all cancers in the body. For certain breast cancers, sure. Ovarian cancer? Probably. Thyroid cancer? Perhaps. But I am not so sure about all the other types of human cancers out there. I suspect if all Americans were Iodine sufficient we'd still have a very high incidence of (skin, colon, lung, ...) cancer due to other diet/lifestyle/environmental factors (junk food, smoking, drinking, pollution, ...). But at least with iodine we can guard against some forms of cancer.