If air-cooling proves sufficient for much of the running time (such as for an aero-engine in flight, or a motorcycle in motion), then oil cooling is an ideal way to cope with those times when extra cooling is needed (such as an aero-engine taxiing before take-off, or a motorcycle in a city traffic jam).
Air-cooled aviation engines may be subject to "shock cooling" when descending from cruising altitude prior to landing.
During descent, very little power is needed, so the engine is throttled back and thereby develops much less heat than when maintaining altitude.
[1] The Wankel engine features oil cooling in addition to liquid-cooling to successfully manage its excessive heat.
Some slow-turning early engines would have a "splashing spoon" beneath the big end of the connecting rod.