Some salts, like sea salts, do contain other minerals, although whether they're at nutritionally significant levels is of course another question. And adding iodine to sea salt wouldn't remove whatever other compounds were in it. Wikipedia has this:

Unrefined sea salt contains small amounts of magnesium and calcium halides and sulfates, traces of algal products, salt-resistant bacteria and sediment particles. The calcium and magnesium salts confer a faintly bitter overtone... Algal products contribute a mildly "fishy" or "sea-air" odour, the latter from organobromine compounds. Sediments, the proportion of which varies with the source, give the salt a dull grey appearance. Since taste and aroma compounds are often detectable by humans in minute concentrations, sea salt may have a more complex flavor than pure sodium chloride when sprinkled on top of food.

Source: https://en.wikipedia.org/wiki/Salt