I think the whole framing of climate issues as "saving the planet" is misguided. Yes, the planet will remain livable pretty much no matter what. The threat of climate change isn't the literal destruction of the planet, or even the literal extinction of humanity. Rather it's such severe changes to the planet's biosphere that human civilization would be threatened.

Potential climate catastrophe scenarios where rapidly rising sea levels submerge many inhabited areas and various knock-on effects disrupt the food chain, even in those scenarios the planet is fine on geological timescales. What's not fine is human civilization, on human timescales. And I think this is what gets inadvertently hidden behind planet-saving rhetoric. If the worst case scenarios are "only" about billions of people dying due to resource scarcity and conflicts over said resources, this is something to be avoided.

I see a parallel with global nuclear war. Such a war would probably not end the species, there would be groups of survivors here and there. But human civilization as we know it would be over, with the global population being back to what it was in the hunter-gatherer era.