Seriously amusing to read. Not my experience at all. Did your brother remove the innermost plate? I don't see it in the photo. Maybe best not to ask.... (ask me why I know what that last little leftover o-ring is for....)
I have now rebuilt two pumps. Its easy breezy.
Since the pump is coming out anyway, if the line is not coming off easily, remove the pump first and the line later.
To remove the gear nut get two people. One holds the gear in heavy leather gloves while the other rattles off the nut with an impact. To remove the gear, use a rag to protect the teeth from the puller.
What is the "rear pump ring body"? Of its the outer spring wire that holds the back plate on (by the screwdriver tip in your photo), compress the back of PS pump with a large c-clamp and then its pretty easy to pick out. The "machined parts" comment makes me think its something else and inside - like the dark metal part by the red pen circle in your photo. To get that out I tapped the pump open-side down on the wooden workbench a few times till it and the vanes (inside their ring) slid out. Then, to get the last part out - similar to the dark metal part by the red pen circle in your photo - after the driveshaft was out I slammed it down on the same bench and it slid out easily.
The rubber band around the vanes is a great idea. I found them mostly stayed in due to the viscosity of my PS fluid. As long as one stays in you are fine because you can look at the orientation and wear marks on that one and put the rest back in the same way
Dave & Chris - lets get together and knock em out.