The fact that every discipline uses the same test for admissions says more about the quality of academia than the scores.
That is not necessarily true. Its a testament to how education is treated as a commodity by College Board and ETS. Academic programs don't care what your scores are. They only use them to help you get fellowships so they don't have to pay for your salary.
The real shame with the GRE is that it says nothing about the quality of the student. I knew someone who just failed their qualifying exam and is getting kicked out, even though he scored a PERFECT on the GRE and had external funding.
The GRE is a pretty generalized test. Most undergrad programs use either the generic SAT or ACT as their test factor - it's a efficient and useful way to measure a large group of people. The most important prerequisite for most graduate programs is an undergraduate degree in that field. How you performed (as well as the quality of the program you graduated from) is likely the most useful tool, so I can see how the GRE would be downplayed.
In some graduate programs, specifically law and business, what field your undergraduate degree is in isn't that important. I'm guessing this is why the LSAT and GMAT tend to weigh more in admission decisions than the GRE does.
What discipline is completely without use for reading skill, or for the kind of secondary school mathematics that appears on the GRE quantitative section?
The GRE doesn't measure reading skill, that's what the NALS is for. The GRE verbal is just a test of how many wank words you've memorized. Same goes for the math test.
Evidence for this? (By which I mean, do you have any evidence that the GRE verbal section has failed validation studies to which it has been subjected by graduate schools?)