You're vastly overestimating the impact of such an eruption on a global scale. Yellowstone already has erupted in the past. In fact, in some cases so recently that humans were on the planet and larger mega eruptions when hominids were around. They've actually found geologic evidence of the ash zone. It literally covers like half the US and was over a meter thick hundreds of miles from the location. But keep in mind that it didn't cause any massive ELE of note. It's not like 90% of mammals or whatever went extinct.
Here's a historical context
Answers to questions about volcanism at Yellowstone.
www.usgs.gov
Sulfur does fuck up the atmosphere to be sure. In fact, that is partially why the non-avian dinosaur killing asteroid was so bad...it struck directly on a location that released a FUCKLOAD of sulfur and put it high into the atmosphere. My old paleontology professor also theorized that it may have struck in the Spring due to studies of plant fossils, because the super acid rain would have fucked up plants bad and because they used all their energy to emerge for Spring, they were less able to recover in the subsequent years (which in turn fucked up the food chain and why pretty much most land species weighing more than 50 lbs went extinct).
The Siberia Magmatraps are also theorized to have heavily caused the Permian-Triassic extinction, in conjunction with asteroid impacts. That extinct event was one of the worst in history...like 90% of all life died on the planet.