If Harrison Bergeron wanted to make a political statement, he would have made his point and then attempted to escape. But his decision to dance with the ballerina until he was shot and killed makes it clear he had no intention of leaving. He just wanted to die free.
Harrison's non-attempt to escape fits with the story's themes of knowledge and ignorance and the desire for freedom. While Harrison shows knowledge of himself and of the societal rules and the desire to take away the freedom to be different, he also seems to be aware that any political statement he makes will not be heard by a society that has forced ignorance.
Harrison's actions are not about changing the society, but liberating himself from both the literal and figurative chains of America's forced equality in the year 2081.