alt Hacker NewsThe difference in cooling potential between cold water and water at ambient temperature is minimal. Cooling with water primarily comes from phase change or heat exchange; both can move vastly more heat than a small difference in temperature.
Chilling the water would massively complicate the logistics with a very marginal improvement in heat removal.
Heat exchange is proportional to the difference in temperature though (in reference to your "or heat exchange"). Colder water would cool faster. The tank isn't at boiling either, so it's not like you'll be able to phase change away a bunch of energy. I guess you'll still get some evaporative cooling, but there's a limit to how much you'll get just from the ambient temperature (the exterior of the tank is relatively cool, presumably because the "gummed up" interior is inhibiting heat transfer)