Again, watch your gauges. Quality engine oils are developed by chemists, and most require being 210-230 degrees to operate effectively, and efficiently.
You should at least have these numbers to allow the condensation that develops in your oil to evaporate. If not, the condensation (water) turns to an acid like mixture, that corrodes. Nothing wrong with a cooler...if you need one.
From my C5 Z06 autocross days......you will see a 15-20 degree oil temp spike from extended high RPM operation on a 40-60 second autocross lap in a LS motor......run the cooler!
On a STREET / DD car -- if I ran an oil cooler (air to oil version) I'd run a thermostat in the system to make certain the oil gets up to temp. Running the oil too cool could lead to moisture building up. I don't know if you'd need that in a water to oil cooler as then the temp would be more controlled.
Like Charlie said -- we have to tape over our air to oil coolers at the track to get some heat in the system. But those are race/track cars... and are driven hard for 20 minute periods much of which is at WOT. Imagine now just running around on the street... and all the stopping etc that doesn't create any heat.