If you were to say that Yellowstone erupting leads to World War III, well, I wouldn't argue that it's implausible. But, similarly, world wars are not extinction-level events. Nuclear weapons can't cause extinction: nuclear winter is basically volcanic winter with a much less effective stratospheric pump (a VEI6 eruption, such as Krakatoa, pushes at least 1km³ of rock into the air; a VEI8 eruption, like Yellowstone, is 100× that). There's a hard cap on the proportion of your population you can mobilize to front-line troops, and while it is possible for one side's losses to exceed this fraction, on a global scale, only 60% or so of the population can be killed in a war, a far cry from the 99.9999% you'd need to kill to effect extinction.
It's certainly possible for an astonishingly large proportion of the population to perish, but extinction requires extermination to be nearly complete.
I think your expectations are overly optimistic. Perhaps it wouldn't be 99.9999%, but in the event of global nuclear war, supply chains for virtually everything (including food) will come to a complete halt. People in cities will be utterly screwed when fuel and food run out.
People in farmland might not be that much better off. Masses of starving, potentially-armed people searching for food will not necessarily be the most respectful of people's property rights. We can expect animal wildlife populations to plummet as they're killed for food, and agricultural output to drop precipitously as supply chains for fuel, fertilizer, and non-terminator-gene seeds go to zero, not to mention as it becomes exceedingly difficult to protect large patches of farmland from raids.
We have exhausted enough of the easily-accessible resources close to the Earth's surface that all it takes is a sufficient dip in technology, and we enter a rut which we can never get back out of.
It's certainly possible for an astonishingly large proportion of the population to perish, but extinction requires extermination to be nearly complete.