Biological weapons can probably do this 10x faster and via weaponized viruses meaning no ICBM's needed, just a good enough virus to be carried by human and animal hosts that infect the world. Or infect common crops and watch the world burn as everyone starves.
A total war between the US/NATO and USSR sans nuclear weapons would also probably have ended western civilization and probably drawn in everyone else too. World-killer? Maybe, maybe not, but the world left over would have been one no one wold want to live in. The same way a post-nuclear world would be.
If anything nukes kept US/NATO and the USSR from going hot. That conflict could very much been a world-ender and was itching to happen for decades.