...The only tutorials I can find show this cross shaft being replaced with the Jeep body completely removed.
Can this part be removed & replaced without taking the body off?
What ever you watched to illustrate removing these parts was probably much easier to film with the tub off. Not easy showing how - with everything in place. It will come apart though. Most diagrams are in Section J of the SM. As posted, remove the upper pads/shanks first. The clutch arm is the only one clamped to the shaft. Yes! Yes! Yes! Take a picture (or several) before you start!
I wouldn't be surprised that your shaft will look pretty good after you get it out. The bracket (and the tube) might have some wear on it and allow the clutch arm to rock side to side. The brake pedal arm itself (since it slips over the shaft) will also be worn. I think this happens when the jeep isn't lubed enough. I had a machine shop install a bushing on the brake arm to tighten it up (it's getting close to needing replacement).