Thyroid cancer has an extremely high survival rate. Doctors would argue it is because they remove the thyroid before the cancer can spread to other areas (like lymph nodes). Since people can survive w/o a thyroid I can see where there is a tendency to dismiss it as a nuisance (, like tonsils, appendix). But in this day and age of litigation a doctor is pretty much forced to advise on thyroid removal in most cases. To suggest taking massive doses of Iodine and returning in 90 days to see if the cancer is gone is risky (in terms of potentially getting sued).
If thyroid cancer is a slow grower, like with some forms of prostate cancer, then I can see where trying iodine/selenium (for example) would be prudent. However I don't know how virulent thyroid cancers are.
Of course it's best to prevent thyroid cancer by nourishing the thyroid with Iodine and selenium and making the right dietary/lifestyle choices ... especially if one has Hashimoto's. Clearly thyroid cancer is FAR more prevalent in America than it should be.