How would they process into human-usable formats and get it distributed in time in the initial period after the nuclear exchange? Beans are easy, but I would imagine a large proportion of people don't even own a mortar and pestle or other means of improvising a flour mill. It is certainly possible to improvise one, but access to information would also be disrupted immediately after a nuclear war so they couldn't just fire up youtube and watch one of these Primitive Technology videos on how to make a mill. We have forgotten the old ways and I fear many people would simply not be able to cope in a harsher environment without modern conveniences and comforts.
I'm not concerned about food shortages causing human extinction as such. Supermarkets with just-in-time logistics leave no buffer room while the feed is processed, but humans can survive 10's of days or longer without food (as long as they're hydrated, with some variability for body weight, access to vitamin supplements, and so on) and outside of major population centres (which are probably craters by this stage) you'd imagine a good part of the population have decent food stockpiles, and at least some farms would hopefully not be too affected by fallout and could continue producing enough food for at least a small village's worth of people. IIRC you need ~150 genetically healthy breeding pairs of humans to have enough diversity (assuming careful management to avoid inbreeding and so on) to viably repopulate the earth, which should be reasonable - although getting them all into one place might be harder.
Rather, I worry food riots and the associated issues such as blocked roads, torched buildings, looting, and so on could push a civilisation that is already severely damaged over the edge into total collapse.