If you set the carbon tax at about $1/gallon of gasoline, the corresponding carbon rebate would be about $1000 per family per year.

That wouldn't affect rich people much; neither the $1/gallon nor the $1000 extra income is significant. But many rich people get rich by being penny-wise, so many would change behaviour, by buying an EV or similar.

But for poor people both $1/gallon and $1000 per year is significant. If gas was $1/gallon more expensive, poor people definitely would drive less.