Iodine is often contraindicated in many cases in which Hashimoto's is present as it can make an autoimmune attack worse than it might otherwise be. That said, you don't mention which drugs you're referring to that eliminate iodine from the formula, so it's hard to answer your question.
As for CBD for thyroid issues, I don't know of any studies; however, if one is hypothyroid and requires replacement thyroid hormones, the only thing satisfactory treatment is thyroid hormones. CBD is not thyroid hormones so it can not replace them. On the other side of the coin, I've never heard of CBD being able to stop the production of thyroid hormones in the case of hyperthyroidism, either. One needs anti-thyroid medication in order to do that.
CBD oil might be satisfactory to treat other symptoms of a thyroid condition, but if thyroid treatment is adequate, symptoms should be alleviated.