* climate change may lead to a "horrible catastrophe" of unknown substance with probability A
* this "horrible catastrophe" may lead to a nuclear war with probability B
* the nuclear war may lead to humanity's extinction with probability C
* you are claiming that A * B * C is very close to 1
If that's right, I call bullshit. There is a ton of potential triggers for nuclear war besides climate change, we didn't have one yet and I can't see why climate change should be the biggest concern in this regard. Moreover, nuclear war is hardly an existential threat to humanity and may even cool the planet a bit.
It would be horrible, it would be comparable to the largest mass killings of 20th century, but it wouldn't be anything that different from what we have survived in the past and it wouldn't be a "near extinction event". "Just" a horrible catastrophe.
It is debatable how close we have been, during various past crisis, to an actual large-scale nuclear conflict. I don't think we can deny our raw capability to cause a near-extinction-level with our existing technology (nuclear, biological or otherwise). Our political and military ecosystems, flawed as they may be, have kept that capability in check.
But would such a "horrible catastrophe" stress that ecosystem way past the point where it will remain confidently under control? Absolutely yes, I believe that beyond any reasonable doubt.
With our global structure out of control, is there is a significant probability for that destructive capability to be put into action? Absolutely yes, I believe that probability to be significant, beyond any reasonable doubt. I don't see how it can't be otherwise.