The biggest problem with this idea is that we don't really know what the costs are going to be.
In practice, poor people would be beneficiaries of a externality-capturing tax, because by and large, rich people use a lot more carbon. More flights, bigger cars, bigger houses, etc.
It becomes a big political fight over how much of the revenue should be directed to social programs, so the tax policy changes every 4 years. A straight rebate gets wide support and taking it away becomes more and more unpopular as people get used to receiving it.
You have a popular bill and you shove something politically unpopular in it, in hopes it will pass anyway.
The purpose of such a tax would be to prevent climate change by discouraging "needless" activities. Want to take your 10th vacation this year by flying 500 miles? Maybe if the ticket was $20 more expensive that prevents some people, at the margin. If such a tax is gradually increased each year and alternatives get cheaper each year (direct carbon capture, electric cars, renewable energy), then eventually a society will be carbon-sustainable with much less harm to the people at the bottom of the ladder.
You were lied to. The best sort of pigovian tax scheme, you collect the tax from those that can pay it. And use the money to help those that can't do what you need them to do.
AKA. Put an excise tax on new gasoline cars. Because someone that can afford a new car can certainly pay the tax. Do the same with business leasing cars. Use the money to fund programs to get poor people out of their gasoline cars and into electric ones.
A gas tax on the other hand will be perceived correctly by poor people as a rubber hose that hurts them a lot. And wealthy not at all.
The idea here is to allow the cost of decisions to be felt in the wallet, but not hurt the spending power of those who are making climate-conscious purchasing choices.