Isn't it wrong to pray for people to be wiped out?
The words are strong. The prayer asks God to uproot, break, bring down, destroy, bring low and subdue. It is fair to ask how that fits with a tradition that cares about every person.
The Yehi Chevod page tells how Rabbi Meir's wife, Beruriah, answered his prayer that the hooligans who tormented him would die. Here is the part that matters for this prayer. She said the verse in Tehillim means that sins should end, not sinners. Then she read the verse's last words, "and the wicked will be no more," as the result: once the sins end, there are no more wicked, because they have returned to the right path. So pray for that. He did, and they changed.
Now look at the Hebrew of this prayer. It says harish'a, "the wickedness," not harsha'im, "the wicked people": "all the wickedness will perish in a moment." That is an observation about the wording. The Gemara does not tie it to Beruriah, so treat it as something to notice, not as a ruling.
But the prayer also names people: "all Your enemies" and "the insolent." For "Your enemies," one explanation is a principle of Rashi (the great medieval commentator on the Torah and Gemara) on Bamidbar 31:3, that someone who stands against Israel is as if he stands against God. Rashi says this about a war with Midian, not about this prayer.
Shmuel HaKatan, the sage credited with writing the blessing, is the one who taught: "When your enemy falls, do not rejoice." The tradition holds both: the harsh words, and a rule against enjoying an enemy's fall.
Berakhot 10a,Tehillim 104:35,Rashi on Bamidbar 31:3,Pirkei Avot 4:19Why would a prayer leader be removed for skipping just this blessing?
For most blessings, a leader who skips one is not removed. When he is reminded, he goes back and says it. This blessing is the exception. The Shulchan Aruch rules that if he skips it completely, he is removed at once, because perhaps he is a heretic. The Gemara states the same worry.
The Yerushalmi lists three blessings like this and names ours by its ending, "who subdues the insolent." The other two are the blessing for reviving the dead and the one for building Jerusalem. The Tur (a law code from the 1300s) gives the reasons. Skipping the first might mean he does not believe in reviving the dead, and skipping the last, that he does not believe the Messiah will come.
Then the Gemara asks about Shmuel HaKatan, who forgot his own blessing and was not removed. Its first answer is that he wrote it, so nobody would suspect him. But maybe he changed his mind since? Abaye says a good person does not turn bad. The Gemara pushes back with a verse in Yechezkel and with Yochanan the High Priest, who served for eighty years and in the end became a Sadducee.
The Gemara's final answer is about starting. The rule applies only to a leader who did not begin the blessing. Shmuel HaKatan had begun it. Rashi explains that he started it and went wrong in the middle. A leader who began it is allowed to finish it. The Shulchan Aruch rules the same way: if the leader began it and made a mistake, he is not removed.
The Meiri (Rabbi Menachem Meiri, Provence, around 1300) adds a practical reason for the long wait. The wording was brand new, and other people did not know it either.
Berakhot 29a,Shulchan Aruch, Orach Chaim 126:1,Yerushalmi, Berakhot 5:3,Tur, Orach Chaim 126,Meiri on Berakhot 28b,Rashi on Berakhot 29a