Normally they do, but when you shut it down you need something to get rid of the decay heat. That's why nuclear power plants have these emergency generators. Those failed at Fukushima, leading to the meltdown.

Some newer designs opt for passive decay heat removal, eg through convection.

In American PWR designs decay heat removal post loss of power is accomplished by natural circulation. Height and temperature differences between the steam generators and the reactor vessel produce flow of primary liquid between the two. Heat is rejected via the secondary system. The limiting factor becomes make up water to the secondary side of the steam generators.

Wouldn't convection also be impacted if the plant is flooded?

They have to shut down during the earthquake to ensure they maintain coolable geometries. Cooling is easier at lower power.

Some Fukushima reactors did have some self-powered safety features, like a little steam turbine that comes from the boiling coolant and is hooked to a coolant pump. These features are at least a little related to why some reactors failed at different times than others in Fukushima.

But seriously, there is enough decay heat to do more of this so it's a pretty good point.

Fort St. Vrain ran the reactor cooling pumps off of process steam, but had lots of problems with steam leaking through the bearings and contaminating the coolant. Electric pumps don't have that problem.

https://en.wikipedia.org/wiki/Fort_St._Vrain_Generating_Stat...

Running some kind of heat engine off decay heat should be possible, but it would be very different from the primary turbines, (You probably couldn't use it during normal operating temperatures, but it'd still have to be connected to the coolant circuit) have to be nuclear rated, (expensive!) and only be used in an accident scenario where you lose external power and the diesel generators are then destroyed, or run out of fuel.

That is the basic idea behind the test that the Chernobyl reactor was doing during the accident.