This is a common misunderstanding, I think. But it actually still works. What matters is that the tax makes fossil fuels expensive relative to other energy sources. To take the extreme examples to make it clear: if you go 100 off fossil fuels, you pay no tax but get the dividend, so you win. On the other hand, if you were to for whatever reason choose the maximally carbon-intensive lifestyle you could, you'd pay way more in tax than the average dividend.
In other words, the tax you pay varies with your carbon emissions, while the dividend is constant (or rather, depends only on the total emissions of everyone). You still have a very strong incentive to reduce emissions, as it would reduce your living costs substatially but reduce your dividend by only a fraction of a penny (I.e. the change in total tax collected divided by the population).
So the 9.1% inflation we just experienced did more for the environment than an 8% carbon tax? Am I understanding this correctly? Did the emission of carbon actually go down because of inflation?
I think, to the degree that inflation is driven by rising gas prices, yes, it'll have a pretty substantial effect on reducing emissions. But my understanding is that inflation is also driven by things like federal reserve policy, stimulus bills, and supply issues due to the pandemic. Even then, if people are just consuming less and doing less fossil-fuel intensive stuff due to feeling squeezed economically, that'll reduce emissions. So it's kinda complicated. In the short term, it really depends on so many factors that I couldn't tell you, though I bet someone well versed in this sort of thing might be able to give a rough answer.
But also consider: in the long term, higher gas prices due to supply shortages encourage companies to drill for and produce more oil -- because they are getting the profits. But higher gas prices due to a carbon tax have the opposite effect -- the buyer is paying more, but that's because the producer is paying the tax, and passing on some or most of that cost to the buyer. They aren't making more profit, but less -- reducing their incentive to pull more oil out of the ground. So that's something powerful that a carbon tax does that inflation doesn't.
And inflation is (ideally) a temporary thing, and variable. A carbon tax is intended to be permanent (and ideally, slowly and steadily increasing), something predictable and inevitable that people can plan around. You'll act differently if you think gas is going to go back down by 20% than if you think it's going to stay at the higher prices forever, or keep getting more and more expensive over time.
My understanding is that even the possibility of a future carbon tax has caused some companies to start reducing their fossil fuel dependency, because if a tax happens, they'll be in a better place than if they did business as usual.
Inflation, recessions/depressions, and other economic maladies do indeed suppress carbon emissions, but I'd hope you'd agree that a more precise and direct lever (like a carbon tax) would be vastly preferable.
In other words, the tax you pay varies with your carbon emissions, while the dividend is constant (or rather, depends only on the total emissions of everyone). You still have a very strong incentive to reduce emissions, as it would reduce your living costs substatially but reduce your dividend by only a fraction of a penny (I.e. the change in total tax collected divided by the population).