Have the Iodine doctors (Flechas, ..) responded to your paper? I think you are questioning these doctors' methods in a very professional manner. Hopefully these fine doctors will respond in a like manner.
The Iodine doctors are pushing for Iodine load testing because people like to look at numbers to see where they are relative to a target. I think there is little disagreement that most everyone is deficient in iodine. The big disagreement is how much iodine to take to reach sufficiency, ... and how does one know when sufficiency is reached. The iodine load test sounds arbitrary to me, but that's all there is. It might not serve much value at all, and it seems most on this forum have not taken it.
I think it's best to judge matters based on clinical symptoms. For example healthy thyroid function, no FBD or PCOS, and a healthy prostate *probably* indicates an iodine level that is not *seriously* deficient (, although perhaps not optimal). But should such folks supplement with iodine anyway? And if so, by how much?