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.