With a clean coolant system, the OEM oil coolers are fine. The problem is that yours looks like it is pretty dirty. A dirty coolant system is hard to clean up and you risk plugging up a new oil cooler if you can't get it completely clean. Most Techs recommend chemically cleaning (using Cummins Filtration brand chemicals - Restore Plus and Restore) when the system is very fouled/plugged.
https://www.cumminsfiltration.com/sites/default/files/lt36625.pdf
Ford sells a product similar to Restore Plus.
Before doing that, you might try backflushing the oil cooler. Closely inspect what you backflush out. It helps to know if it is gel or solids. The best way to backflush is with a system like that sold by fixur6.
https://fixur6.com/
https://www.youtube.com/watch?v=yIX1rr8zSTs
https://www.youtube.com/watch?v=Va6K3S9-vWQ
If/when you decide to chemically clean:
The Restore Plus is an acidic cleaner and for iron scale and mineral deposits. The Restore is for silica gel and even helps if oil contaminated. You typically want to try cleaning the system with the contaminated oil cooler in place. The problem with that is that it has a bypass in it if the differential pressure across it is too great. If the cooler is bypassing, it won't get cleaned. Every system behaves a little bit differently. Cleaning it can get laborious.
If I had the money, at least with a severely plugged oil cooler, I would install the BulletProofDiesel air cooled oil cooler. If that is too much money, then installing their remote located oil cooler will help in that it can be changed out more easily (and cheaply) if it gets plugged again.
Be aware, the coolant system capacity is 7 gallons. The BEST you can do when draining is to get out half of that. That even assumes you drain coolant from the block plugs. This means that a lot of the old stuff stays in. That is why a back flush is very helpful.
Lots of people flush with hose water at first with the last few flushes being distilled water. This works ok, but IMO you need at least 2 distilled water flushes. When you go to add coolant, you need to add 3.5 gallons of concentrated EC-1 rated ELC coolant, then top off with distilled water. It is best to use a vacuum fill tool, but it isn't absolutely necessary.
