Yes, if there is an underlying deeper problem, supplements aren't the answer.

However, if it is possible to become slightly iodine deficient by eating a modern diet in the US-- which seems plausible-- and if slight iodine deficiency causes or contributes to mild depression -- which also seems plausible -- an iodine supplement could be a easy solution.

> the underlying problem, which is usually emotional trauma

This is untrue. The emotional trauma hypothesis is thoroughly debunked and has been for very many years. The vast majority of people with depression do not have any emotional trauma.

If you attempt to reduce an extremely complex system of incidental traumas, re-enforcements, psycho-social effects and a multitude of many other factors down to "there was a death in the family" or "my father hit me once" then I would totally agree with you, that has been thoroughly debunked. There are healthy people whose fathers hit them, or have had deaths in their families, this is absolutely true. However, you do your mind (and yourself, obviously) an extreme disservice if you think of its mechanics in such simple terms (especially if you suffer from depression.)

It's like troubleshooting a computer by attempting to discern its vague history and comparing it to other computers, and then deciding that it's a hardware problem because some computers have similar software and function okay. Any computer technician can tell you that modern PCs are way too complex to repair using such simple logic (unless you just start formatting drives.) Now, there may indeed be a hardware problem, but you can't discount the OS based on a three-point checklist.

I grew up in a family with an extremely negative view of people who had mental health problems, and when I came down with depression I ran to fibromyalgia, chronic fatigue, etc. because I didn't want to be "crazy". But all I did was set myself up for worse problems down the road when that rationalisation wore thin. Now, I'm not saying that taking an emotion-centred approach will solve all of your problems and make your depression go away -- it probably won't. But it will help you cope. Numerous studies show that it helps people cope. How could that be, if your thoughts didn't matter?

I didn't say that thoughts don't matter. I'm not sure if you're aware that I strongly advocate for CBT.

But CBT doesn't aim to uncover emotional trauma. You talk about an emotion that is causing you distress; the hot thought that's associated with that thought; and then evidence for that.