Does car heater use gas to generate heat?

The water pump is a simple centrifugal pump. A centrifugal pump at a given rpm uses less power with less flow (as would result from increased drag).

I would agree with you if the thermostat is restricting flow into the pump but not if it is restricting flow out of the pump. A restriction on the intake would lower the water pressure, on the output it would raise it.

But I do believe that some cars have the thermostat on the intake side of the pump now so that is a good point.