Yes, oil rigs could dredge up nutrients at less cost than outfitting ships to dump iron into the sea. The drawback is where the oil rigs are is not where the adding of iron would have the positive effect on algae. The North Sea and Gulf of Mexico are already nutrient rich and suffering from ocean dead zones where so much algae is blooming that oxygen is consumed, maybe toxins, and most marine life as we know it dies. Nitrogen is a more damaging pollutant of ocean than is plastic because it is a nutrient that fuels harmful algal bloom that kills fish. In Boston Harbor hundreds of striped bass chased thousands of menhaden up the Mystic River into a hypoxic ocean dead zone and all fish died to litter the shore. The fish kill was caused by too much nutrients (N) from off the land, resulting in too much blooming algae.
In Iceland, where geo-thermal heats their buildings and moving petroleum around for cars and trucks is a pain, they have large vats of algae on the land. The algae is grown in warm waters with nutrients. Then it is processed into fuel for their cars and trucks, much like ethanol. Instead of drilling for oil, oil companies could grow algae and process into fuel on site next to the fuel pumps. Photosynthesizing algae pulls about 3 tons of C02 out of the air for every ton of carbon biomass. As a result, for every ton of fuel used by your favorite internal combustion machine, you would pay for the net draw down of two tons of CO2 out of the atmosphere. Maybe refining algae into fuel costs less than drilling for and refining petroleum, and oil company profits could go up.