Would the risk of goiter be eliminated by addition of iodine?

No: At least not in his country where iodine deficiency would be rare.

It depends : If you grew up in the United States, it is unlikely that low iodine is the cause of goiter, since so much of our processed food is made with supplemental iodized salt. However, if you are from an iodine-poor area of the world (central europe or africa, for example), then the addition of iodine may help.