Avimelech....The thyroid is certainly important to the body. But at any point in time, a healthy thyroid will store 50mgs iodine whereas the rest of the body stores or contains 1500 mgs iodine. So its blindingly obvious to me that the rest of the body needs iodine just as much as the thyroid does.
Here is a doc I wrote some years ago on the whole body benefits of iodine in the body -- and the list in the doc is by no means definitive or complete;
It's always better having someone who is thyroid competent at your side monitoring your progress who can advise you directly. You might also be helped with bio-identical or natural thyroid hormones. You should avoid all synthetic thyroid hormones from the regular doctors -- they will do you no good at all and generally tend to make things worse.
To help you pinpoint your thyroid problem you should really do the correct lab work. Curing thyroid problems isn't really as simple as just taking iodine. You will need to supplement the right minerals and vitamins to enable the supplemented iodine in your body to be correctly absorbed, transported, stored and used properly in the body. That's why you should find a thyroid -literate, iodine-literate doctor in your own area to help you if you can.