Why does disk loading mean anything when rotors/propellers are not disks? "The helicopter rotor is modeled as an infinitely thin disc with an infinite number of blades that induce a constant pressure jump over the disk area and along the axis of rotation."

This is spectacularly not true of common propeller disks. Most of the area in propeller/rotor disk is empty space, through which air flows freely from top to bottom. 

So why do we take this mostly empty disk and use it's area in aerodynamic calculations, when it would seem that we should rather be using the area of the blades themselves? 

It seems to permit "fooling" disk loading equations by, for example, decreasing the blade chord without increasing the span, but still getting the same results from equations. So, what a I missing?
The disc model is used wherever generalities of behaviour are studied without the need to know individual blade characteristics. It captures the mathematics of those generalities in a simple way that leaves out extraneous detail which would make the maths impossibly laborious.

Typically the aircraft designer will use the disc model in the design of their airframe, leaving the blade details for the propeller or rotor designer to figure out.