It just sounds a little dry. I think environmentalists have done a good job drilling the fact that it is a bad thing into us, but in the face of it “climate change” sounds like a sort of generic and neutral process. Global warming was a little better (I think? Or maybe I just grew up knowing it was a bad thing).
Climate Instability sounds sort of clearly bad (I think most people consider instability bad).
If we're workshopping this for you: "climate change" has alliteration and is only 3 syllables compared to "instability" which has a mouthful of 5 syllables.
It’s just the way the conversation happened to flow. I think everyone has an opinion on names.
For solutions I’m in favor of really aggressive carbon taxes and sanctions/secondary sanctions against that don’t implement them. But that’s the sort of thing better left to regulators and diplomats.