You're 100% right, except for one detail: human politics. Unfortunately, the organizations here on Earth actually capable of solving the climate change problem

You're 100% right, except for one detail: human politics. Unfortunately, the organizations here on Earth actually capable of solving the climate change problem (namely, governments) move too slowly and/or are horribly divided against themselves (see: USA). As an engineer, I look at the problem and I come to your same conclusion; It would be far, far more efficient just to save Earth than to try to colonize and terraform Mars. Unfortunately, Earth politics being what they are (hell, USA has > 500,000 deaths due to COVID and STILL around 45% of the population thinks it's fake or no worse than the flu), our only realistic option for ensuring the continuity of the species is to colonize another planet so that if the governments on Earth collectively decide to ensure its doom, there is another place where humans can continue the species. It's sad really. But it's the reality we're dealing with.