Why some universities don't require GRE AWA section?
It's only a guess but as an admission committee member I have weighted AWA lower than the other two as well for the following reasons:


The range of the score is pretty narrow (0 to 6 with 0.5 increment) and according to the guideline, it appears a good tool to find out who can't write, but not who can write well. Anything about 4 is kind of hard to tell.
Verbal and Quantitative have algorithm that is carried out by machine. AWA is graded by hand, thus can make admission committee feeling doubtful about the objectiveness or specificity to their program ("GRE's thoughtfulness is not our kind of thoughtfulness!!"). And I'm not to say algorithm is perfect, but it may still beat grading by hand (for instance, you can get locked into a low-score path just because you got the first question wrong, making the first few questions more determining than the rest.)
Most of the writing skill and thought process can be gleaned from the personal statement. (Of course, conspiracists can also argue that the statement may be written by someone else. But they should also know that all GRE writing topics are publicly available (1, 2). Two of them will be randomly selected and presented in the test. In a way, the students could have also gotten external help on this part prior to the test as well.)