The reason 4 strokes run oil coolers is because they run the same motor oil over and over. They're especially prevalent on air cooled 4 strokes because the oil actually does a lot of the cooling (oil squirters in the bottom end and usually the heads are designed so that the oil bathes the exhaust port and valve guides to keep them cooler)
Motor oil is good up to about 375° but then it begins to break down (the hydrocarbon chains begin to oxidize and break into shorter less lubricating chains which is what "coke" is) If air cooled 4 stroke engines didn't have oil coolers the motor oil would burn up in short order and the crankshaft, piston, and camshaft would be toast.
Typical 2 strokes, use total loss oiling. Oil is either injected into the intake tract or mixed with the fuel and once it has done it's job it's burnt and ejected out the exhaust pipe.
The transmission fluid (typically motor oil but can also be modified gear oil or ATF) in a 2 stroke is not subjected to the same heat and contamination as it is in a 4 stroke. The transmission fluid gets hot, but not nearly as hot because it's not directly exposed to the combustion process.
If you were making a drag bike where transmission fluid temperature could be a concern, you could install a small "spoon" pickup in the clutch cover on the outside of the clutch ring gear to act as a catcher for forcing transmission fluid up and around a cooling loop. Because it would be pressurized the cooling loop would have to stay down low near the footpeg and then dump back in at the bottom of the clutch cover or transmission case.