[jrapdx3]

AFAIK cationic calcium, et. al., combine with dissolved CO2 forming carbonates which having low water solubility eventually precipitate out of solution. Area with "hard water" often have issues with buildup of scale in water pipes. The scale is mainly calcium carbonate and sulfate.

This is normally a gradual process requiring ample moisture. Releasing finely ground rock into the air wouldn't likely be effective. For one thing, such dispersal would be rather dilute. IOW the dispersion wouldn't reliably react with CO3 ions before settling out of the air. Another consideration is the salicaceous content of the rock being a potential health hazard.

The original idea of adding crushed rock to water-containing soil is logically the best way for the project to accomplish its goals.