I don't think that argument fails, generally. The availability of rice, beans, and potatoes in bulk quantities on Amazon still does not make them accessible to the poor. The poor are frequently unbanked, and prepaid card schemes with predatory fees are not an improvement. They also are more likely to live in neighborhoods where packages are stolen.

Apart from that, if people eat nothing but the foods that can be stored unrefrigerated for long periods, they're going to end up with health problems, even some of the same ones caused by processed foods.