Radiator in front?

You get minimal cooling of the engine block from the air around it so, the thought "hot air around the thing you want to cool" is irrelevant. Besides, the engine block will still run at the coolant temperature, controlled by the thermostat.

There would be an advantage in keeping heat away from other underhood components, especially plastic parts. Also, in keeping the heat away from the passenger compartment, at least in summertime. However, the later gets most heat from the exhaust pipes, which would still be there.

Problem is how could you plumb such a large flow of air around the engine, and why do so. My guess is that the engine bay would run hotter without airflow from the radiator. It would still have a large heat input from the exhaust. Your experiment seems to have already been done in rear-engined cars. Not an expert, but seems I have seen many on TV that have a radiator which vents hot air outside, not thru the engine bay. I recall reading that those cars have problems with high engine bay temperatures.