Getting to space is absolutely NOT a solution to climate change. We have to solve climate change SO THAT we can continue on our path to become space faring. If we can't manage to survive on an Earth decimated by climate change, then we obviously have no chance in space or any other planet in this solar system.
We 100% will go extinct eventually if we don't expand beyond Earth. And my point is that if climate change destroys our civilization now and we get flung back into a new dark age, I don't think it's likely that we'll ever manage to spread beyond Earth.
Climate change is an existential threat to our species because it threatens our ability to become space faring.