Literally true. However:
- If it takes them 10 minutes to fire up the generator, then 5 minutes to restart the network and registers, that is no big issue (in a many-hour outage)
- At least in my part of the USA, many supermarkets do have generators - because storm damage causes local outages relatively often, and they'd lose a lot of money if they couldn't keep their freezers and refrigerators powered. Since the power requirements of the lighting and registers are just (compared to the cooling equipment) a rounding error, those are also on generators.
Seen it done in USA, for a Target next to a Kroger grocer. Kroger lost everything that needed cold after reserve either ran out or wouldn't start, but Target had a contingencies contract and lost no product.
But both major supermarkets nearby worked on diesel generators and payment by card worked flawlessly. I guess they had satellite connection.
It might have been more complicated in small villages but people living in rural areas ually still use a lot of cash.