In any case, besides addressing the root cause vs. proximal cause, you couldn't even address the proximal cause anyways: it's more likely that the world could do something about climate change than about war in general.
My take is that if you remove nuclear/biological war out of the picture somehow, climate change is not an existential risk. If you remove the latter then the former is still an existential risk (and there are unfortunately a lot of other possible sources of geopolitical instability). So the fundamental source of risk if the former. But it's a matter of opinion.
Conventional or chemical warfare, even on a global scale, are definitely not existential risks though. And like I said, probably not even nuclear. Biological... that I could see leading to extinction.
Still, like I said somewhere else, even all out nuclear war is unlikely to lead to human extinction. It could push us back to the neolithic in some scenarios, but even then there is some disagreement about what would happen.
Of course, even in the most optimistic scenario it would be really bad and we should do everything we can to avoid it - that goes without saying.