The thermostatic control valve, then the fan, then the core. The valve diverts oil around the cooler until working temperature is reached, and if it sticks open the oil never reaches the cooler at all - so the system overheats with a perfectly good cooler and a running fan.

The fan and its drive. An outright failure is obvious, but a motor running slow or intermittently produces the same gradual temperature rise as a blocked core and is much easier to miss. Hydraulic fan motors on mobile equipment fail in the same gradual way.

Usually, and it should be tried first. Chaff, dust and insects pack the fins until air cannot pass, and careful cleaning from the clean side outward restores performance that looked like a failed cooler. Bent fins can be combed straight, and both are far cheaper than a new core.

Bypasses the cooler while the oil is cold, so the system reaches working temperature quickly instead of being cooled from cold start. It then progressively directs oil through the cooler. Stuck open it causes overheating; stuck closed it means the oil never warms up properly.

Tube bundles, tube sheets, gaskets and sacrificial anodes. The anode is the item most often neglected - it is there to corrode instead of the cooler, and once consumed the cooler itself starts corroding, so replacing it on schedule protects an expensive component cheaply.

Yes - a fan guard keeps hands and debris out of a rotating fan, and a missing one is both a hazard and a route for damage to the fan itself. It is a cheap part that is frequently left off after maintenance and should be treated as part of the cooler rather than an accessory.

By elimination - a working thermostatic valve, a fan running at full speed, and a clean core, with oil temperature still rising. At that point the core or tube bundle has genuinely lost capacity, usually through internal fouling, and replacement is justified rather than speculative.