So, what is wrong with the statement?
While the statement is not false, it has a kind of type error. Iodine is not in a category that can or cannot be synthesized in organisms (or any chemical reactions). We only talk about molecules in this category: e.g. Vitamin C can be synthesized in some organisms, but not others.
The following is also not a false statement: household waste isn't handled by the JVM garbage collector. However, at least we know it was written in jest. To nitpick that one would be to betray not getting the computer science dad joke.
At worst, it implies that, like essential amino acids, it CAN be synthesized by some living organism.
the fact it doesn't add more, possibly irrelevant, information does not diminish its usefulness.
it does not imply anything about things that aren't being discussed. it's abnormal to read everything with a qualification like "in our bodies" as a implication of some other hidden meaning.
as technical people we probably enjoy delving into tangents and pedantry far too much. but, for example, adding a statement "... In fact, iodine is an element and cannot be synthesized ... " would be a tangent and should be avoided or edited out.
It is absolutely normal to read every condition given in some technical subject matter as being relevant, and decent writing satisfies this.
It takes extraordinary effort to maintain a constant suspicion that every stated condition might be irrelevant to the proposition to which it is attached.
this is a magazine article, not a proof. so i'm certain the author felt it was relevant to say it that way because the whole article is about what happens in the body, whether or not it was logically relevant to the condition.
the fact that he made a narrower than necessary true statement does not imply he meant anything beyond that either, even if you read it pedantically. it takes extraordinary effort to maintain a constant suspicion that every unstated condition might be relevant too.
It does not imply that.
Rhetorically, it does have that interpretation.
When some unconditional proposition is presented together with some red herring conditions, a less informed reader may take the interpretation that the conditions are relevant; i.e. that it is falsified when the stated conditions do not hold. Because, why would the presenting expert include irrelevant conditions?
"I've noticed that Smith doesn't beat his wife in public."
That doesn't logically imply Smith beats his wife, but he does have to be beating his wife for the "in public" condition to be relevant. To assume that the condition is relevant is to assume that Smith does beat his wife. If the condition is not relevant, why is it there?
I'd like to be able to assume that all conditions given in serious writing about STEM subjects are relevant, so I don't have to waste my time suspecting they are not.