The oil cooler is just a lump of metal, but can fail. the only sure-fire way to tell is to remove, clean (easier said than done, try a parts washer) and then inspect/pressure test.
99% of the time, just replacing the O-rings (FORD ONLY) and cleaning everything really well solves the problem.
Re; white smoke on cold starts - this can be from a tune, or weak injector, or sometimes even lack of HPOP
output. A buzz test/contribution test will tell you a LOT. I like to start with the easy/cheap, so I would say test all your GP's and relay first. If GP system is greenlight, then go for buzz/cont test.
Let us know what you find, there is also an additive called ArchOil that is pretty damn amazing on injectors, a fresh oil change and a bottle of that stuff made a huge difference on my 7.3, and their fuel additive is well worth the money. Modern diesel sucks, and needs some love to make it right.
Re: mechanics - yes, hard to trust, but if they will show you whats wrong, give a realistic explanation, and aren't trying to sell you a bucket of prop-wash, they might be OK. Diesel engines are specialized knowledge, and PSD
engines are further so. Find a shop that works on them. Stay the hell out of your local napa/gas station service center. I do about 80% of my own engine work on this truck, its not hard, you just have to remember 3 rules about diesels:
Keep em clean and it'll run forever. Neglect it and it's fubar.