The engine generates heat during operation. No operation, no additional heat. The water/coolant helps dissipate the engine heat as the heat transfers from the engine to the water and the water circulates. The water goes into the radiator where the heat from the water is transferred and the water is cooled (sometimes with the aid of the aux fan).

When the engine is shut off the water stops circulating so the water increases in temperature and continues to increase until it matches the engine temp and then they both decrease in temperature together to reach ambient temperature.

The aux fan comes on at a certain water temperature among other reasons. So it comes on when the engine is started after it has recently been allowed to increase in temperature due to water/coolant not circulating.

All this to say that you will need an electric water pump to solve your perceived issue. Our cars were engineered so that we do not need to do anything like that.