Yup, I'd say you did a pretty good job explaining the cause and the symptom/effect.

Repairing a toasted axle for a few hundred seems a bit fishy. From the sounds of it I'd bet that the diff was beyond saving (hot enough to melt = heat distortion, pitting/warping of bearing surfaces, fusing of components to housing, etc.) and the entire axle would need to be replaced, which would be at least a few hundred for the part and even more for the labor.

I can kind of see the front axle being smoked from turning two different size tires, although the spider gears should save the diff (they're there so that the two tires can turn at different speeds while turning a corner or when one doesn't have traction) and transfer that stress to the t-case which would fail due to the front and rear shafts turning at different speeds.

Looking at nothing but the rear diff, what could cause it to get so hot that it melts the bushings? Already low fluid would be my first guess, Occam's Razor and all... could have had a leak and this just happened to be its time to go. Hard to say what killed it without seeing it.