[Quickurt]

Actually the size of the water cooling system is calculated to keep the coolant at a usable temperature. It carries off heat from much hotter areas to be dissiminated into the air through the heat exchangers (radiators). The coolant is simply the vehicle used to transfer internal heat to the air. The oil does the same, and in fact in the air cooled engine oil was the only vehicle. Air cooled the cylinder sleeves directly through cast fins, but the heads were cooled by oil, as well as the internals of the crankcase. The air cooled engine also ran much hotter, overall, and was one reason why they had oil seapages. They expanded and contracted more as the temps rose higher and dropped more with the higher operating temperature and oil seal material had to withstand higher operating and aging temps.
As LB points out above, the oil is also cooled via the water in liquid to liquid heat exchangers, so the coolant acts as the vehicle to carry the heat absorbed by the oil out into the air.
I don't know if any are still used, but some industrial and ship engines used steam as the coolant because it allowed higher operating temps than when the coolant could not rise above the boiling point. The reason cooling systems are pressurized is the raise the boiling temp of the coolant (water).
By the way, a simple physics question. If the engine and the coolant are of uniform temperature, how is heat absorbed from the engine, into the coolant, and then released into the air through the radiators, to return to the engine and absorb more heat?