As a n of 1 that doesn't fit your theory: I have been dealing from iron deficiency anemia for a few years. This certainly has not improved my condition! Just yesterday, I finally received an infusion of IV iron dextran (I am no longer able to absorb oral iron due to GI issues). It has not made me worse (yet, and please knock on wood!), and I am certainly hoping that my energy levels with increase over the next few months.
Many years ago, my mother (who was always skeptical about my illness, but wanted to understand) suggested I get tested for hemachromatosis, so I did. I turned out to be heterozygous for this genetics for this, but never had any evidence of iron overload. Then, in recent years, I have been mostly in a low to very low iron state.
Which is not to say that iron overload may be the answer for a subset of people, and this is definitely one of the diseases that should be ruled out in patients with symptoms suggestive of ME/CFS.