Why doesn't Hazel need a handicap?
There are no handicaps on Hazel because she represents 2081 America's idea of equality. More accurately, she is clearly Kurt Vonnegut's example of the lowest common denominator that everyone must be handicapped to become like. She is the reason why people, like her husband George and her son Harrison, must be handicapped.
In literature, characters can be symbols, or stand for something larger. Hazel is such a symbol. She stands for the idea that a normal person is someone incapable of extended thought. This idea of remembering ideas only for an instant ("she couldn't think about anything except in short bursts") suggests a larger desired forgetfulness that many in the ruling class, like Diana Moon Glampers, would prefer from their subjects. 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