What have I done wrong here? Can I make the one I ordered work with the grille and headlight parts I have ordered from the same vendor?
From what I see: You haven't done anything wrong. The core support you ordered is designed to be used on '73-80 trucks. However, there were several changes from 1973-1980.
As you have noted these differences, your new core support will need to be modified to accomodate your specific year model. It's not quite a 100% bolt in swap, but close to it.
Specifically, the '73-'74 year model trucks had different grille mounting bracket arrangements than the later years. There were 5 brackets used on the '73-'74 trucks to mount the grille. Of the 5, two were integral to the core support and are located on either side of the radiator opening; the remainder can be unbolted and swapped to your new support. If you're using an original '73-'74 grille, the center and outer brackets are usually sufficient for support. It appears that the body shop has already looked at swapping the outer brackets and marked bolt locations.
There were also differences in the way the air conditioner lines ran through the support, which will need modifications to accomodate your specific setup. Additionally, the upper core support panel and hood latch will need to be swapped (which it appears has been done).
As for the headlights; I'm not sure what you have or the issue they're having- but the headlight parts (buckets, mounts, adjusters) should also swap from your '74 core support to the new support.
Mounting the core support to the rest of the front sheetmetal can be a chore. All I can say from my experiences is: have patience, clean the threads in the bolt holes and bolts, lightly oil the bolts and have several alignment tools to help line up the bolt holes.
Looking at the photo of your original core support, it appears that it's rust free and only marginally pushed back. Could the shop simply pull the original back into alignment?
You may also search the forum for 73 grille or 73 grill and core support.
I'm sure others may have more to offer.