Brian
They do, but the rub is that they still need to be connected to the grid.
The output for a US "Nuke" plant is 345,000 volts and 100's of megawatts, which is connected to the grid in such a manner that all of the turbines on the grid are in sync. A power plant by itself is too small a load by itself to run electrically when presented to the main generator output, and still maintain thermal efficiency of the reactor itself, so the power is sent out and back, so to speak. The power for the pump motors is 6.9KV, and 4160, so the voltage has to be stepped down for motor operation. It's pretty involved actually
In the event of grid instability, which happened at Fukushima, the plant disconnects itself from the grid by virtue of a turbine trip, once the generator output breakers are opened.
Once disconnected from the grid, and in the absence of power on the grid, the back up generators start to power the electrical busses required for safe shutdown and continuous plant operation to maintain reactor core integrity. It is a layered approach based on redundancy for safety margin.
These backup power supplies, diesels, at US plants are tested, and surveilled, continuously. The inability to demonstrate that back up power systems are available 24/7 will force the owner/operator of a plant to shut down, administratively, to meet regulatory requirements. The diesels are a very big deal at US plants.