"Carbon storage" would normally refer to physically sequestering purified carbon dioxide or other carbon bearing compounds. I agree that storage of that type is not practical.
Enhanced silicate weathering is IMO the process with the best prospects for large scale atmospheric carbon dioxide removal. It is relatively slow but the thermodynamics are favorable and the kinetics are still orders of magnitude faster than waiting for unaided nature to restore the pre-industrial equilibrium. Enhanced weathering CDR just accelerates the kinetics of the natural chemical reaction that turns alkaline silicate rocks and CO2 into silica and alkaline carbonates. Doing it on a scale large enough to make a difference would be a gargantuan undertaking, of course, because the scale of the problem is also gargantuan.