From chemistry point suggested process is indistinguishable from the following process: pass air over CaO or Ca(OH)2 solution to turn it into CaCO3. Heating of CaCO3 will release CO2 thus regenerating CaO, which could be reused again. This process would require energy input — like MIT tech.
Excess of CO2 in atmosphere is not necessarily bad thing. More CO2 in atmosphere means more carbon will be available for capture by plants, which means more crops and trees.