A Gray sister in her first administrative position is already climbing her steps in the Tower hierarchy, while a Red training to fight other channelers would always have her political ambitions hit the ultimate glass ceiling, the fact that a Red sister has virtually no chance of becoming Amyrlin. Only the Blue has had more success in raising Amyrlins than the Gray.
Personal power outside the Tower is hardly interesting to any AS. They're all rich, anyway, as respected as any monarch, really, and deeply loyal to the Tower. When Gray sisters negotiate, they increase the Tower's influence, not their own (say, affecting how laws are interpreted and border disputes settled). Though Temaile saw negotiations as just a suitable outlet to exercise her sadistic side, as she was said to treat negotiations as games to control and hurt people.