I think it is safe to say the SOHC4 engine wasn't designed to have an oil cooler. And, as it was to be sold throughout the world, expected to encounter high temperatures. So, the cooling system's capability/capacity was incorporated to allow both the oil and the engine to survive in those high temperatures under routine use.
As the system can handle hot extremes, yet has no regulation ability, at normal and cold temps the system actually over cools both the engine and the oil. Anyone who has found the brown "mayonaise" (oil/water emulsion) in the oil supply has seen the proof of over cooling, as the oil did not get hot enough to evaporate and expel the water.
Now, if the power density of the stock engine has been increased. And/or the engine is operated extensively at near peak power output as well as high temperatures, it is possible to overtax the stock cooling system, and toast the oil supply. Then, if the damaged oil is not replaced, engine damage can result. For this reason, race engines get an external oil cooler, to save the oil (nee engine) during the extremely abusive environment it experiences, well outside what normal street driving entails.
Because a race engine benefits from an oil cooler does NOT mean it is also beneficial for a bike operated on the street. You cannot legally operate in race mode for very long, even if you have made yourself an "assault bike". At normal street use power levels, the stock cooling system STILL provides proper cooling of components and oil.
To wit. I have NEVER seen nor read of any engine failure in this forum where oil over temp was the determined cause of a failure. Oil starvation, yes, overcooked oil, no. And yes, I have overcooked the oil in my 550 due to partial starvation, high environmental temperatures, and excessive street idling. A change in oil was all that was required to return the bike to regular service.
Ok, so you choose not to believe me and put on an oil cooler anyway. What's the harm?
If you use an oil cooler system without a thermostat on a standard use bike, the oil becomes over cooled and can't flow throughout the engine as the engine designers planned. Will it immediately blow up? No. Probably not. Will wear factors increase, I believe so and you have shortened the service life of the engine. Further, the normal condensation that occurs within the engine at each cool down cycle, is at risk of not reaching a temperature to expel it during run time. Expect brown mayonaise, and note that lubricating with water is not nearly as effective as actual oil throughout the engine. If water build up is not properly addressed promtly, water presence instigates a biological action where amino acids form, which far from lubricating, actually eats away metal inside the engine. This also shortens the service life of the engine.
If you use an oil cooler system with a thermostat on a standard use bike, there is a high risk that the thermostat will never or very seldom open. This allows the oil in the cooler to stagnate, collect water, and build up amino acids. Then, if or when conditions are actually encountered where the thermostat opens, all that contaminated oil w/acid contained in the oil cooler finds each and every bearing journal in the engine to munch on. Shortened service life.
For a standard bike, operated in standard or median environmental conditions, the oil temp can easily never open an oil coolers thermostat, (how would you know if the stat operated?) allowing the oil within the cooler to not circulate.
Ok, so you modded your engine for increased HP, and you are worried about the increased heat waste from the extra HP. This is a valid concern IF you plan to use the extra HP most of the time. It still only takes 10-20 HP to cruise on the freeway no matter what your engine's maximum capability is. The stock cooling system can handle that easily, without any need for an added oil cooler.
I'll inject here that an oil cooler adds another level of complexity to oil changes, as some method of draining it should also be addressed.
In my opinion, the lamest excuse for adding an oil cooler is sitting/idling in hot traffic.
If the air can't get around the engine to cool it, how is an oil cooler radiator going to get cooler air through it replacing the heated air?
If you expect that little cooler to take the heat away from the engine, it needs the same oil movement volume as the air flow to be effective. The oil has to absorb the heat inside the engine, be replaced, and moved to the oil cooler where in can give up its absorbed heat to air... that is not moving. Has anyone noticed that the oil pressure and oil movement volume is lowest in the engine at idle speeds? The SOHC4 just is NOT an oil cooled engine, that is a tertiary function of the oil in this motor. Further, heat energy only moves when there is a differential between thermal masses. If the air next to a 300 degree cooling fin is also 300 degrees, the air absorbs no heat, and the cooling fin remains at 300 degrees. 300 degree oil on the other side of the cooling fin also remains at 300 degrees. That, my friends, is thermodynamics/physics. This is one reason why nearly all internal combustion machines that survive routinely during prolonged stationary events have fans, whether there is an oil cooler on it, or not.
And that, people, is why I believe an oil cooler on a street use SOHC4 is an affectation, rather than a functional improvement. And no one in this thread has yet shown a verifiable demonstrated need, rather than rationalized assumptions.