While his nerves should be super damaged, my headcanoned explanation for this is that Joshua is a ghoul; turned right after the fall and that is the only reason he lived. Here's a couple points for this:
1. Though questionably canon, Van Buren notes that the grand canyon lost its status as a national park so people could start mining uranium there, which left it very radioactive. (The loss of the lands protected status might also explain why the Sierra Madre was able to be built so close to the canyon.)
He would've basically been covered in burns, fell from an extreme height, and then exposed to high levels of radiation, ouch! Perhaps he even fell into the water and it got into all his wounds, quickly irradiating him.
Even if he didn't get submerged in the water, the ambient rads + eventually having to drink the contaminated water would eventually ghoulify most anyways.
2. (Most) Ghouls are notoriously covered chronic wounds don't heal. While they can regenerate from a variety of injuries, the ghoul body will always prioritize healing the nervous system first, while the rest of the body only regenerates the bare minimum to keep itself somewhat functional.
Why fix a tear in the skin that doesn't risk infection instead of maintaining the brain or restoring the ability to feel?
My point is, Joshua is practically covered in chronic wounds, looking closer to a marked man than the average ghoul.
His burns are the chronic wounds. His nerves were originally destroyed by the fire. But after becoming a ghoul, the nerves regenerated. Everything damaged by the burns are unhealed except the nerves. He can feel all his forever-injuries perfectly fine.
Technically, if this is true, he wouldn't need to bandage himself (at least for the reason he specifies he does it), since ghouls are immune to most infection. Though it would likely be more comfortable to cover the wounds with a bandage barrier and to keep dust and dirt out of them, so him doing so is by no means unreasonable.
3. "Why's his voice sound normal though?"
The only main two ghouls (that I can think of) that don't have any of the typical raspiness are Cooper Howard (an actor) and Dean Domino (a singer). Both had careers that required vocal training and strength, and I theorize this is a big factor as to why their vocal cords are seemingly fine. They simply know how to not ruin their voices and preserve them since it was very necessary for both their respective careers.
Joshua Graham was an interpreter. Another role that requires training in vocal range, warmups, and the preservation of ones voice.
My point is, ghouls who have specifically worked with their voices prior to ghoulification, and know specifically how to take care of them, seem to keep their normal voices.
4. "Why doesn't he just say he's a ghoul?"
Dunno, Joshua is a weirdo. Denial. Not realizing due to the fact he's different looking than ghouls he's known. Weird moral thing. To avoid the stigma of being a ghoul. Something like that. Being covered in chronic burn injuries and being a ghoul are functionally not all that different.
Tldr, my theory is that Joshua is a ghoul with a lot of chronic wounds. I've seen Ghoul!Joshua theories before, but I just wanted to throw my two cents in, especially in regards to his chronic pain.
While a theory I think is fun, it probably isn't canon. Forgoing the ghoul stuff, @mojavediscodancer 's thoughts about healing wrong/having a variety of injuries/psychosomatic pain is likely the canon answer!
