I made the proposal specifically to sequester carbon in calcium carbonate as a sophomore taking college chemistry and oceanology (a great course).
It was an obvious thing to do.
Though sewer sludge would provide some nitrate and phosphate, and you could enrich it with a little bit of iron, the vast size of that oceanic region means that the scale of the amount you'd have to ship out there is really large.
We make a lot of sludge near coastal cities and are looking for better things to do with it. I'd bet there's a fair amount of iron in it as well as trace minerals that we probably don't know about. The transportation by water would be cheaper than we now use by land and in most cases the sludge could be piped instead of burning methane to dry it. I like the containment aspects of the gyre. The stuff goes straight down.
Thinking as a futurist, the idea has merit. I'd have to worry about trace metals that might be harmful (like arsenic and chromium and lead) that might be in the sludge, and we all know about the problems of mercury and cadmium in the food chain; but I wonder if you could process the stuff with metal-specific chelates that would leave the iron in it and take out the more insidious stuff? If you could finish with an organic-rich, iron-enriched mixture, even if 98% of it went to the bottom (no big deal, that's a lot of sea floor), the other 2% might be enough to get a sustainable bloom going. If the economic and enviromental circumstances made the climate favorable, who knows?