Yes, you get the money back eventually, but there's a time lag and no amortisation. Big purchases might become impractical.
Like I said, it's a fool's errand to try and fix climate change and wealth inequality in one fell swoop. Sadly, trying that will just end up solving neither.
Rich people are rich, they can do more than poor people, now that means they can buy a fancy car and good healthcare while poor people can't, soon it might mean they can fly across the globe four times a year while poor people can't. We should fix wealth inequality, too.
If you give more money to the less well off, they'll just polute/consume more. Meanwhile the well offs will still consume as much and just pay a bit more for it.
Or you make it potent enough, but then it will only effectively affect the less well off. Unlikely to be popular.
Ultimately, it feels like you need a non-linear carbon tax, but it sounds unfeasible to implement.
But hopefully the extra income will soften the blow. The idea being that with it, you can either afford some of the high-emission-behavior you were used to. Or you can afford more of the low-emission-behavior than you were used to. Comparatively a lot more, because it's both taxed lower and you have extra income.
Although having a carbon tax that's redistributed equally per capita (so you get a monthly check) is an interesting idea that ends up reducing emissions (because it disincentivises them) and is mostly progressive (because rich people consume and emit more).