Assessing a tax doesn't actually clean the air. All it does is ensure that only the rich have the right to fuck everyone else over. If the government committed to funding carbon capture at a rate of at least as much as they would've in the counterfactual where there was instead a ban plus the additional amount funded by the carbon tax revenue, taxing the externality would probably be adequate redress, but that isn't ever going to happen.
You have to both put the entirety of the tax towards it and ensure that the fact that you're funding it through the tax doesn't decrease the amount of additional funding you put towards it or else that decrease effectively decreases the tax rate (at least as far as carbon capture is concerned). The second part is harder than the first.
This is what happened when lotteries started spreading across the US. The proceeds were earmarked for schools, so naturally, the property tax rates (which pays for most school funding) was held in check, resulting in no actual increase in school spending.
If we can offset "the expense of everyone else" (in this case via a carbon tax), why not?