It should be much more cost-effective to float wind turbines to pump surface water to benthic depths. Then, the whole ocean is your CO2 collection surface. You might also pump water from (lesser) depths to the surface to distribute over coral reefs to relieve both heat and pH. (Picture a line of these some miles seaward, all up and down the Great Barrier Reef.)

The wind turbine nacelle could be substantially cheaper than in the typical electrical generating turbine, instead coupling mechanically or hydraulically to the water pump.

Some people worry that this would saturate the deep ocean with CO2. A bit of calculation shows this is an idle concern: the shallow ocean is already being saturated with CO2, with immediately dire consequences, and there is overwhelmingly more deep ocean water than surface water. So long as atmospheric CO2 falls off over the coming century, there would no long-term problem.

Geoengineering that tackles the root problem, CO2, is fundamentally different from schemes that only (e.g.) try to block insolation. Other engineering is going on to tackle upstream CO2 emission, but we have a huge stock of CO2 already built up in the atmosphere that will need to be drawn down.