"Has anyone estimated how much carbon the various flood basalts and basalt traps could sequester?" Yes sort of. It is huge but limited by surface area of rocks. Crush them to dust and they get very frisky. Especially buried in soil where CO2 concentrations are high. Olivine is the friskiest of all, but when it is 'carbonated' it releases heavy metals that are not welcome additions to soil.
CO2 injection for enhanced oil recovery does not stay down there. At least I don't know that it has been shown to stay down there.
Did I miss some disappointing followups to this?: A step forward for carbon storage | PriusChat One of the basalt storage projects is in Iceland. Another was at Wallula, along the Columbia River in my state, run by the Pacific Northwest National Laboratory. I don't see the capture method in Iceland as being globally practical, at least not yet, in part due to its energy cost. But the storage and mineralization portion seemed quite promising, at least when I last heard anything. My state has a very large inventory of basalt, but the planet has plenty more and larger flood basalt traps available.
I was referring to CC&S in the US. Currently, it is mostly used for oil recovery. https://sgp.fas.org/crs/misc/R44902.pdf An example project was a coal plant capturing 4,717 metric tons of CO2 a day, which was used to boost an oil field's daily output by 14,700 barrels. That extra oil is the equivalent of 6,321 metric tons of CO2. Greenhouse Gases Equivalencies Calculator - Calculations and References | US EPA Counting CO2 used for carbonating water for CC&S(some source might do so) is actually more productive. In better news, it looks like future US projects will be more true, geological sequestering.
This seems to be the state of the art: "Potential for large-scale CO2 removal via enhanced rock weathering with croplands" https://doi.org/10.1038/s41586-020-2448-9 There's plenty of basalt. At $180 per ton of CO2 sequestered, may not be enough financial interest. At highest implementation, one runs out of croplands to bury the stuff. Which is rather amazing all by itself.