Carbon pricing is not intended to redistribute wealth.
Then what's the fucking point. Wealth distribution is core of all problems in society.
we also don't have evidence of this.
Wrong. You can literally do it with your own taxes. Go look up the corporate taxes for any public company (they're public so they publish those numbers), and figure out what percentage of that company's total wealth it is. Then compare that with how much tax you paid as a percentage of your total wealth.
You can even do it with billionaires because some of them, like musk, actually voluntarily share that info. Last year I paid over 20% of my total wealth in income tax alone, musk paid less 4%.