Why isn't a turbine turning to provide power to pump the water in the reactors?

Another example of how we ignore simple solutions and waste available energy.

So, they build these plants to produce electricity, but then they have diesel and batteries for backup power? If it's producing heat, it's producing steam, which means it can turn a turbine. Even with the reactor shut down, as long as there's heat, there should always be a turbine turning to provide the energy to pump the water and run equipment. Why would you do anything else? This way there would always power to operate the plant in an emergency.

This is worse than the coal mines just venting out the methane from the mines rather than capturing it or using it to power equipment in the mine, because it could prevent a disaster.