I think there is difficulty in separating thyroid health (that is, having enough selenium, Iodine and other nutrients for robust operation) from Hashimoto's. Hashimoto's is an autoimmune disease whereby antibodies stupidly attack the thyroid. As with most (all?) autoimmune diseases there are both genetic and environmental factors in the disease being formed, and most (all?) autoimmune diseases are incurable (, although remission is certain possible).
Okay, so once one has Hashimoto's the trick is to get the body to stop generating anti-thyroid antibodies. I think discovering and eliminating the aforementioned environmental variables are key. For many going on a gluten free diet has helped tremendously in calming Hashimoto's. Other dietary changes might help too. Fighting back pathogens (bacteria, yeast, virus) might help also.
Now what about iodine? Well Iodine is a super antibiotic/antifungal monster, so it might tamp down on an autoimmune response. I think also by making the thyroid healthier it would delay inevitable damage caused by anti-thyroid antibodies. So I think Iodine (with selenium) for Hashimoto's sufferers make sense. I can easily understand where folks supplementing with iodine can reduce/eliminate their T4/T3 hormone meds.
Lastly, I don't see how iodine deficiency is any sort of cause of Hashimoto's .... but I am not an iodine/thyroid expert. Therefore supplementing with iodine, while perhaps helpful (as I mentioned above), is nothing close to a Hashimoto's cure. [Certainly if I had Hashimoto's I would give iodine/selenium supplementation a try ... despite what my doctor would say! :)]