That part about funding unrelated agendas is partially why many leading economists recommend returning the revenue as a flat dividend. The revenue from the carbon tax is just divvied up and given back to everyone as a monthly direct deposit. https://econstatement.org

This also addresses the other concerns you pointed out below. The dividend helps to both (a) allow you to invest in improving your energy efficiency or (b) soften the blow where you can't. You may remember the Yellow Vest movement when France tried to apply a carbon tax. If France had given money back to the people, they probably would have had a lot more success with it.