Evenly distributing the revenue assumes the effect of the negative externality (climate change) is evenly distributed. But climate change tends to harm poorer people more than richer people, so it makes sense to distribute more revenue to them.
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.