It's not 1945 anymore, when there was a sharp divide between urban areas and rural areas. The majority of the U.S. population lives in suburbs, and many only go into cities for cultural events. The suburbs are already wired into the U.S. logistic network - that's where are Walmarts and Amazon warehouses are - and are decentralized enough that they're not likely to get taken out by an initial strike.
Coronavirus (particularly Omicron) was much more devastating to our logistic network than nuclear war would be, because it affected essentially all truckers and port workers at once.
I'd encourage you to read the pamphlet; it's put out by scientists whose full-time job was to plan for the continuation American society in the aftermath of a nuclear war.
I think you're incredibly optimistic about the robustness of logistic networks in the aftermath of the complete destruction of all of the major cities in the United States. You think people are going to bother coming into work to stock food on shelves, deliver it, etc in the immediate aftermath of a large-scale nuclear exchange? This isn't even getting into the immediate effects of say, all of the forests in the western US turning into firestorms.