Regarding your high psi oil pump. You really need to hook to a computer and find out what your pressure are.There is a required press. and actual press. If it’s your pump I would think it would act up on a cold or hot condition.With the correct software you can dead head the pump to see what it is capable of putting out. The 6.0L needs something like a minimum of 800PSI to fire the injectors.Sounds more like an IPR Valve (Injection control pressure regulator)If you want to screw with it yourself,you could start by removing the IPR and if the screen is torn/ripped then replace it.Good luck trying to remove it since it’s hard to get to.This engine uses high press. oil to fire the injectors. I wouldn’t just throw parts at it unless your loaded$.Also use the approved filters for this engine (per ford/international)since some of the other brands of oil filters/elements don’t stay primed and the other brands of fuel filters don’t have a patented auto air bleed, so the others will only use half of the filter since the other half is filled with air and can’t be purged. Don’t try to save $ on filters.