Diesels don’t like sucking air- not even a little bit.
on an '04 with a 6.0L though, I would stop going to the dealer and see if you can find someone local that has a lot of experience with these. The 6.0’s are a known Ford/International disaster, and pretty much killed the decades long relationship between the two companies (or started the downfall at least.)
Have you checked the Fuel Injector Control Module? The FICM takes an ungoldly amount of voltage (48 volts, if I remember correctly,) and will start acting up if they drop even a little bit. The solder joints within them were known for coming loose with years of use and heat. Usually they result in more of a hard start than a start and die, but things happen, ya know? lol