I can not say for certain why the scientists/engineers of the time did not pursue the use of metallic sodium for engine coolant, but it was most likely due to the high melting point of 208°F/98°C.
In order to use a liquid to cool an engine it has to be able to move fairly quickly. Since sodium would be solid when the engine was first started, the various parts of the engine to be cooled would have to reach the melting point of sodium before the sodium could begin to move. The problem with this aspect is that while the sodium near the cylinders might turn to liquid quickly enough to be pumped away (thereby taking the unwanted heat away) the sodium a little bit further away would still be solid. The solid sodium would of course block the flow of the melted sodium. This means the entire engine would have to be above 208°F/98°C before the sodium could begin to flow through the engine. The problem also applies to the sodium in the coolant radiator and transport hoses. The heat from the cylinders would have to be conducted through the liquid sodium in the engine to the solid sodium in the transport hoses and the radiator and in turn melt the sodium in the radiator. This would take a significant amount of time, too long for the survival of conventional engine lubricating oil. (In general the temperature around the cylinders has to be kept well below 575°F/300°C so the lubricating oil does not boil in the transport channels, reducing/eliminating the lubricating value.)
A method for gradual heating up of the engine to operating temperatures would be doable (I think) but would have to be more sophisticated & complex (I think) than the relatively simple methods used in the water/glycol cooled engine. Also, a method of controlling the start-stop of the coolant pump would have to be implemented, different than the simple PTO used on the WWII engines, so that the coolant pump did not try to turn before the sodium melted throughout the entire system and was able to flow, and would stop the pump turning before the sodium returned to solid form again. Effectively monitoring/controlling a system like this would (I think) have been problematic with WWII technology.
The second part of the problem that I see is that while water/glycol coolant is liquid at room temperatures and remains so at temperatures down to about -50°F/-45°C, metallic sodium would begin to freeze whenever the temperature anywhere in the system went below 208°F/98°C. This means that the sodium cooled engine would have to be run at much higher temperatures (say about 250°F/121°C minimum to say about 500°F/260°C maximum) than the water/glycol engines, in order to prevent cold-spots and sudden blockage. This would require the materials used in the engine to be able to withstand the higher operating temperatures, so probably no plastic or rubber seals, and a different type of lubricating oil.
Disassembly, maintenance, and repair would also be significantly more difficult than for a water/glycol system.
I am not familiar enough with the chemistry of sodium and other materials what are commonly used in engines to comment on corrosion and such.