Carbon taxes do not pay the external costs of carbon emissions. Full stop.
But that was never the intent of a carbon tax, was it? It's not a reparations program.
The purpose of a carbon tax is to make carbon emitting-technologies more expensive, so that the market will be encouraged to find alternatives that emit less carbon.
Without that, it's difficult for alternative technologies to get a foothold in the market, as they are forced to compete with carbon-based technologies that are allowed to pollute "for free", thus neutralizing (from a monetary perspective) the advantage of the non-polluting technologies.