These concerns are currently being voiced by former Nuclear Regulatory Commission Chairman Dr. Shirley Ann Jackson. The reactor trips and the plant stops producing electricity. But, because the grid has failed, there is no offsite source of power to operate the equipment necessary for bringing the reactor to a safe shutdown.
Nuclear plants rely upon their own diesel generators in such a situation.
However, these generators are unreliable. For example, just a few years ago Three Mile Island's diesel generators were inoperable for three months and nobody knew it. Even without diesel generators, coolant can be circulated by steam-powered pumps. A bank of batteries supply power to the control room and the valves that will permit that operation. The batteries will only last four hours.