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.
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.
It's terribly scientifically inaccurate, though. Airburst nukes (the type that would generally be used on cities) do not generally spread large amounts of fallout into the atmosphere; they do not generally spread large amounts of fallout at all, because the fireball doesn't reach the ground. We'd likely get the sort of massive urban blast damage from the first half of the movie, but wouldn't get the nuclear winter, collapse of crop harvests, and widespread starvation from the second half of the movie. And we should expect to see local government spring up in areas undamaged by the blast - if the national government falls, state/provincial or local government will step into that void, because power abhors a vacuum.
In the book Sapiens by Yuval Noah Harari - he even suggested that hunter gatherers had a far better quality of life compared to someone living in a farming society. Obviously what we have now is far better than either.
http://oism.org/nwss/nwss.pdf
Basically - you're only toast if you're within the fireball, radiation, and moderate blast damage radius. If you're inside the thermal radiation or light blast damage radius, you should immediately take shelter away from windows and face away from the flash, much as you would in an earthquake. Hiroshima is filled with anecdotes of people being vaporized and casting shadows on the wall behind them, while people in the next room over survived with only minor cuts and burns. You have about 5-20 seconds between seeing the flash and the blast wave getting to you; the thermal pulse itself can last for up to 40 seconds, so the less time you spend in it the less badly you'll be burned.
You can view the blast radius for common warheads on NukeMap:
https://nuclearsecrecy.com/nukemap/
Topol ICBMs are 800kt and Bulava SLBMs are 100-150kt; you can expect Russia to use SLBMs on strategic military targets and ICBMs on cities. A hypothetical attack on the Bay Area that targets SFO, OAK, Moffett, Alameda NAS, and the Port of Oakland with Bulavas and downtown San Francisco and San Jose with Topols would leave basically all of the East Bay intact, along with the Sunset/Richmond, San Jose south of 280, and the Peninsula from Mountain View to Hillsborough. It's survivable at the Googleplex, despite being basically in sight of Moffett, as long as you're not near an east-facing window when the blast hits.
In NYC, assuming a Topol on Manhattan and Bulavas on JFK/LGA/EWR, it'd take out most of lower Manhattan but large portions of Queens and virtually all of the Bronx, Brooklyn, and Staten Island would survive.
You do need to prep for an interruption of utilities, but you should do that anyway for hurricane/earthquake/etc preparedness. If you've got ~3 days of water and ~2 weeks of food and are outside a blast radius, there's a good chance of you being fine. Given the large amounts of suburbia that are not in the blast radius of a likely target, food supply chains likely wouldn't be more damaged than they were in the pandemic, and you could go to a local Walmart for food. Depending on how centralized your local water supply is and whether they have backup generators (the electrical grid is a likely casualty), you may keep municipal water as well.