Regarding the Quant Section of the GRE, I believe most of the mathematical concepts that require mastery in order to obtain higher scores become pretty useless to the vast majority of programs once you hit the 50th percentile. I also think that this section favors "traditional" students over those who have taken time off to research, travel, work, etc. since they will have most of these mathematical concepts fresh in their minds (thus, a bit of ageism might be at play). My thinking is that this portion of the test is really more of a test of a student's level of discipline, since a PhD program's curriculum will likely cover most of the statistical methodology that would be considered of value to the research program, and many young researchers will already have exposure to this through publication experience and undergraduate coursework. It's a bit strange to me that the GRE Quant section does not cover any statistical concepts, as I would think these would encompass the most wide ranging applicability (if you truly wanted to accomplish the impossible and apply a "one-size-fits-all" test to every graduate academic program). I'm not sure if theoretical scientists would feel singled out by this since I am not quite familiar with their lack of experimental design, but who cares about them since they are a bunch of weirdos who tend to transcend the status quo anyways.
Concerning the Verbal and Writing portions of the test, research academia does not really require eloquent writing abilities in order to publish significant findings (unless you're writing a review article, maybe?) and some could even say that verbal ability as a factor in publication potential encourages ethnocentric bias in a multicultural academic environment. However, realistically speaking I would probably want one of my students to be able to construct a solid argument without me having to revise their work extensively (although I might also consider this the "fun part" of the publication process since I actually enjoy writing). Again, I might say a good cut-off here is at about the 50th percentile.
I definitely think the GRE can currently help sort out the "numbnuts" factor in a potential PhD candidate, but it's also a bit like asking someone to practice quilting in preparation for a martial arts tournament. In my opinion it is utterly boring and very likely useless. To be fair, I suppose a high score does demonstrate a level of commitment and discipline. However, the reason I am attempting to enter this field is because of my fascination with the subject matter and I would rather spend my free time (which is pretty scarce) on research projects instead of jumping through hoops. Academia probably needs to evolve at some point and adapt different tests that focus on each specific field that fuse quantitative, verbal, and subject-specific based content in a way that is appropriate to every program. That argument might essentially consist of trying to convince ETS to make less money.
Anyhow, if you have been screened by a program because of the GRE (and are stubborn about not re-taking it), maybe you should instead focus on writing an incredible grant/fellowship proposal to excel in a different program or bring it back to the table during the next application cycle. The "brand name" of a school can only go so far, and at some point you will have to realize that the ability to coalesce your expertise, interests, and initiative into significant scientific discovery is far more valuable than just hopping aboard the reputation train of someone else's program. If you manage to excel in this, you can probably wind up directing your own research in a faculty position at one of the institutions that screened you for some stupid test with subject matter that possesses little relevance to your career path. Maybe you'll even help form a prominent new research institute :).