There are no handicaps on Hazel because she represents society's lowest common denominator. Everyone in the society needs to be handicapped in order for people like Hazel not to feel inferior.
In literature, characters can be symbols, or stand for something larger. Without Hazel, the reader does not know to what level people would have to be handicapped. The narrator says, "Hazel had a perfectly average intelligence, which meant she couldn't think about anything except in short bursts." Meanwhile, Hazel's husband George's intelligence is "way above normal," so he received mental handicaps in his ears.
In general, the intellectually and athletically elite tend to be the models for social behavior. In this society where "everyone was finally equal," the "perfectly average" are the ones who are the models.
Without Hazel, how would the readers know what people like George and Harrison were being handicapped to be like?
Overall, the story is about how the ruling classes in society want people to be perfectly average like Hazel. They want citizens to stay within their roles and not challenge the control systems they have put in place—handicaps, in this case.