In nearly every dystopian society, the ultimate goal of the leaders isn't the stated goal, "equality" in this instance. For the most part, the ultimate goal in these worlds is control. Handicapper General Diana Moon Glampers is the one who primarily controls the world by ensuring this "equality."
The majority of the people seem to support the goal of this world as evidenced by Harrison Bergeron's parents, who are both pretty accepting of the world's rules. When George Bergeron states that if he removed his handicaps, then "other people'd get away with it and pretty soon we'd be right back to the dark ages again, with everybody competing against everybody else. You wouldn't like that, would you?" His wife answers, "I'd hate it."
However, the reaction is violent when Harrison, whose individuality is so impressive that he is able to "kiss the ceiling," attempts to challenge the control exhorted over him by the government. Glampers enters the studio where this mini-rebellion is taking place and does not warn Harrison. Instead, she enters the studio "with a double-barrelled ten-gauge shotgun" and shoots Harrison, who is dead "before [he] hit[s] the floor."
The problem with these societies is that without forbidding the populace to express their individuality, they lack control. So the leaders, like Glampers, are oppressive and prohibit original thought. This same idea is seen in other dystopian works like 1984, The Matrix, and The Hunger Games.