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.
In the US, a surprisingly easy way to inadvertently cause mild depression is to decrease intake of iodized salt over time. The "natural" iodine content of many foods (bread, milk) seems to be dropping steadily due to organic and other preparation practices. Kelp-extract supplements with around 200mcg of iodine daily can go a long way towards preventing mild depression associated with borderline iodine deficiency.
> To be clear, I pretty much ruled out celiac long ago, because my symptoms are not _that_ serious.
My symptoms were less than yours (constipation once in a while) but I am 100% confirmed celiac. I had positives on the anti-tissue transglutaminase IgG, IgA tests and endoscopy confirmed extensive damage consistent with years of celiac disease. You should rule out celiac disease with the IgG, IgA tests, followed by endoscopy if positive.
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.