I'm stunned that you think that the Accepted and Aes Sedai tests are necessary to make good Aes Sedai. For god's sake, the three best Aes Sedai around today have not taken the AS test, and have shown them to be far more capable as sisters than most others who've passed the test. Cadsuane and Moiraine, other women who rose above the general uselessness of the AS, learned to be as they were outside the Tower. Siuan herself now realizes that she was too cooped up and sheltered in the Tower. It is when she was outside, in less than her full strength, that she was able to realize her fullest potential as a person.

But even if we were to credit the AS system as useful and good, that still doesn't take away the fact that people like Else can be useful. Else by herself may have made no difference, but if 10 years worth of Novices and Accepted who failed to make it to Aes Sedai were kept back with lower ranks, the Tower would have been stronger, and much more ready for Tarmon Gai'don.

To be fair, it is likely that the decision to throw Else out was Sheriam's not Siuan's. But Siuan allowed it, and failed to see how women like Else could be useful.