I am surprised to learn that automatic Cooper S does not have the oil cooler / heat exchanger on the oil filter housing. I wonder why is that?

 

This is true - I do not know why. I do suspect that if you add the trans oil load to the cooling needs - it is too much. The ‘oil cooler’ dumps heat to the radiator. These really need a separate oil to air unit, but it is a small car, and space is at a premuim