Oil has an operating range. Cooler is not better. Although the exact min temp is hard to find, it's generally accepted to be 180*F (about 82*C). Below that temp, the
oil doesn't reach the proper viscosity, and the additive package doesn't work as well. You end up with a lot of strain on the
oil pump and distributer gear - which can lead to an early failure.
For a pure street car, there's no need for an oil cooler. Complete waste of money, time, space, weight, and complexity.
But, if do have a cooler on a street car, a thermostat is a must.
A thermostat when closed will flow about 10% of the oil through the cooler. As the oil warms up, more and more gets directed to the cooler. That allows all the oil in the system to warm up at the same rate. It also keeps the oil from warming up at all if it's cold enough.
When fully open, about 90% goes through the cooler. That helps reduce pressure drop through the cooler.
If it's a street car that sometimes sees track duty, consider a manual valve that you can open first thing in the morning on track day. But completly close for street use.