In "Harrison Bergeron," why do you think Vonnegut decides to write the dance this way?

There are two dances in this story. The first dance illustrates the status quo—what things are like in this world where artificial constraints are placed on individuals to make sure that nobody stands out. The second demonstrates what the world could be like if individuals were free to develop to their fullest potentials. Vonnegut's choices show a real genius for solving problems inherent in writing a concise, comic story set in an alien society. They permit him to convey crucial information with great brevity and humor, and avoid bogging down the narrative with potentially awkward, intrusive, third person exposition.
At the beginning of the story, we're presented with generalizations about how the society is run. We then get a concrete example of how it works by learning about George's transmitter. The entire story will be told inside George's living room where George is watching TV. How can the author show us what conditions pertain outside that small space? How can he give us an impression of how the police state touches the lives of everyone?
In a masterstroke, Vonnegut lets us watch TV and see the dance through George's eyes. We are provided with a sample of the wider society—the ballerinas—and by eavesdropping on George's thoughts, we get some expository analysis from an insider, a member of the society:                           

He tried to think a little about the ballerinas. They weren't really very good—no better than anybody else would have been, anyway. They were burdened with sashweights and bags of birdshot, and their faces were masked, so that no one, seeing a free and graceful gesture or a pretty face, would feel like something the cat drug in.

In just two sentences, we learn the lengths the government takes to achieve its goal (i.e., making dancers wear bags and masks), and we learn that the measures are very effective (the dancers aren't very good). We also learn the insider's understanding of why everyone should support the government's programs (so ordinary people won't feel like "something the cat drug in").
George is an excellent vehicle for conveying these observations. Vonnegut needs to call our attention to both the external reality of the situation and internal view of reality that the government wants people to adopt. From what we've been told, it sounds like very few people living in this society are capable of this sort of analysis. We're told that George is different—his intelligence is "way above normal"—so it's not jarring or implausible that he might have thoughts that would inform the reader in such a convenient way. The author could have conveyed the information by adopting the sort of third-person, omniscient perspective we see in the first paragraph. Presenting George's perspective helps the author reduce such intrusions, allowing the reader to experience this dystopia in a more direct way.
The second dance, too, functions as a way to convey a crucial point with humor,  brevity, and the absence of an intrusive, moralizing, omniscient voice. If Harrison and the ballerina hadn't engaged in their superhuman dance, Harrison's takeover of the TV station would have seemed merely like a terrorist act, a demonstration of bullying and brute force.
By describing, in a few lines, "an explosion of joy and grace" with dancers leaping "like deer on the moon," Vonnegut permits readers to formulate their own answer to the charge that seeing graceful dancers would make people "feel like something the cat drug in." We are also elevated and transported by great talent.