Also true; iron seeding of the oceans would likely sour the oceans, possibly being detrimental to coral reefs.
Discussion
The oceans have already been screwed over pretty bad in some areas, no need to make it worse if there are other solutions.
Impossible to sour the oceans with iron even if we wanted to - journalists can't math.
And near-impossible to lower productivity of High-Nutrient / Low Chlorophyll waters - they're screwed now literally by definition ("Low Chlorophyll"). Easy to raise them, though, especially if silicic acid is already available.