Aloha,
I've had two TR3s here in Hawaii. The first had a heater that I removed, the second did not have one installed. In both cars I didn't have an engine over heating problem after I did some work. I agree with Andy that the cooling system does work if the engine block is kept clean and the radiator has good thermally efficiency. Sediment tend to collect in the engine block at the rear where the block drain petcock is. Radiator fins need to have good contact with the tubes running between the upper and lower tanks in addition to being capable of holding coolant at operating pressure. The fiberboard radiator ducting is also important to force most of the air entering the grille to pass through the radiator.
The heater can be used as an auxiliary radiator, but as Geo points out the heat is going to the cockpit. Even with the heater valve closed on the engine block, my heater would get hot which is why I removed it. Probably the valve leaked by a bit. In my case the engine did not run any hotter with out a heater. I used brass pipe plugs to block the holes for the heater control valve and on the water pump housing where the heater pipe attached.