If the manufacturer supplies the engine with an oil cooler, it needs an oil cooler. Otherwise not, unless you have made significant power-related modifications. Air-cooled engines are designed to run hotter than water-cooled engines, but the rules for oil temperatures are the same for both engines. YOU may be uncomfortable at the higher engine temperatures but the engine has been designed with adequate materials and thermal expansion clearances to run just fine.
From the engine point-of-view, you should expect to see indicated oil temperatures between the gauge midpoint and three quarter mark with occasional excursions to full scale. That’s how the manufacturer selects the gauge mid-point. If the oil temperature is too cool the oil viscosity is higher so you will loose power due to pumping losses.
From the oil point-of-view, you need to maintain lubrication and a high enough viscosity when hot to produce an adequate oil pressure. If the oil temperature is too hot, the oil oxidizes and breaks-down more quickly.
That’s why you should run a synthetic oil, to give you a safety margin against oil breakdown. A regular oil will begin to lose its film strength at temperatures above 220°F (105°C), while most full-synthetic oils can be safely used at temperatures as high as 300°F (150°C) before lubrication-related damage becomes a concern.
As a rule-of-thumb your oil temperature should be kept below 240ºF if you want to change petroleum-based oil at the recommended intervals. For every ten degrees above 240°F, cut your oil change interval in half.
Race car builders usually design for oil temperatures between 230°F and 260°F in order to get the best power. Above that range, engine reliability and oil life becomes a factor in racing.