The largest and, long-term, most effective carbon sink are phytoplankton. They are responsible for half of all photosynthesis, thus absorbing tremendous quantities of CO2. As they individually die, they over millions of years accumulate thick carbon-filled layers at the bottom of oceans. This is the ultimate origin of oil deposits, by the way.

CO2 from the air ends up being absorbed in the oceans. It's really just about; what is the most efficient way to remove CO2 from the carbon cycle. Currently removing from the air has appeared most promising. Perhaps that will change in time.

At a pH of ~8, the majority of dissolved inorganic carbon (DIC) exists as bicarbonate. Inorganic removal of a mole of HCO3- from surface seawater via precipitation of CaCO3 yields a mole of CO2 back to the atmosphere. Easier to fix CO2 via the classic silicate weathering reaction, which grabs 2 moles of CO2 per mole of silicate weathered (using CaSiO3 as a trivial example). This is the essential feedback from continental weathering: the Earth's thermostat.