The amount of CO2 we can pump underground to form carbonates with basalt rock is a drop in the bucket. Unless we find a catalyst that allows us to economically turn tons of CO2 into an inert (and maybe useful) solid or liquid, this is all going nowhere.
No reason to despair, or to stop working on solutions. But today's CCS tech, and the many variations on it, will not solve our problems.
At the moment, our best bet is improved natural sequestration. There is massive potential in better agricultural methods (cf. silvopasture, perennial crops, Upland rice, microbial farming), better land management, re-establishing grasslands in Siberia (cf. Pleistocene Park), farming in the ocean (cf. marine permaculture, upwelling restoration), etc.