Indeed they could make the difference. Remember you're talking about old cars that might not hold but a few gallons of fluid at most and had simple fans. And back in those days you might overheat an engine in the high temperatures in the desert so water evaporating against the radiator would lower the interior temperature to the engine just enough to not have problems. Also, you could refill the bladder in any stream or other water source as needed.
Today's modern engines have more water jackets, and electric fans to help cooling, plus we have overflow tanks to allow for expansion and contraction of the fluids, back in the early days you might lose your fluid if you let the vehicle overheat.
