Someone told me that I can test my generator water pump by removing the thermostat and running the generator with the radiator cap off to see if the water is "flowing" at a good rate.

I did that today and the water in the radiator was not moving!

Is there anything else besides a water pump that could be causing this?

I priced a water pump for a Yanmar diesel engine - $446

The generator is a 20 KW Kohler.