Society is part human, part infrastructure. You need humans to operate infrastructure, but humans without infrastructure leads to collapse.
Let’s say the nuclear exchange is limited, so limited that it only radiates the docks, highways, and the airports of major nations; All of which were evacuated.
Coffee mostly comes from over seas in America and Russia, so immediately all the of coffee that many people depend on is gone. I won’t go over all the specifics, but this will occur with a large amount of services, some non essential like specialty food products, some essential like overseas car parts and specialty electronics.
Then there are the places that consume more food or water than they produce, places that get money through tourism and trade. These places will fail and the people will need a way.
People will try to preserve order, but this will eventually breakdown. If lack of infrastructure makes it so the people of Maine can’t interact with the people of Arizona in any meaningful way, why should they stay in the same country? The more infrastructure is destroyed the greater this effect.
If things get so bad that everyone is back to sustenance farming or minor maintenance jobs, currency might collapse also. Why sell something with money, which can’t feed your family, when you could trade for something you actually need? If the central government falls then money could be counterfeited or stolen.
This will cause society collapse without killing anyone.