Two of the things you mentioned could be causing the issue. First, the idea that moving the starter relay around helps at times could indicate there is corrosion or a loose connection where it's mounted. Check to make sure none of the pins are pushed in or loose.
Next, the neutral safety switch could be faulty also. Start by checking the wiring and connector there is tight and clean. There are two different transmissions this could have come with, but they are both located on the driver's side of the transmission. (See pics 1 and 2).
As far as the throttle body, that won't prevent it from cranking. Let me know what you find as far as the wiring mentioned.
Here is a link you may find helpful when checking:
Let me know what you find or if you have other questions.
See pics below.
Images (Click to enlarge)
Sunday, January 9th, 2022 AT 8:48 PM