FWIW I think the argument isn't that the net change isn't beneficial for the poor, but that the tax is effectively regressive in proportion to income in that e.g. Jeff Bezos could fly a private jet to and from work every day and still pay less for carbon in proportion to his income than someone poor driving to and from work every day.

So while it is a tax that benefits the poor, the poor are also paying a higher portion of their income to support it while the Jeff Bezoses of the world remain largely unaffected, can use this disparity to further widen the wealth gap and—for any carbon tax rate that the poorest half of the population could at all afford—continue eating their beef and heating their houses for an infinitesimally small portion of their income.

Meanwhile, the Jeff Bezoses of the world probably have the best means available to lower their carbon emissions. Their wealth could be used to provide public transportation and green energy developments, so I believe that there are more efficient ways to reduce carbon emissions than consumption proportional carbon taxation. Progressively factor income and profit into it instead, and lower other income and profit based taxes to simultaneously reward successful businesses/people with small carbon footprints and make those that benefit the most from poor people driving to and from work pay for it.