I'm bothered by the prayer that calls for the destruction of our enemies. What if I don't want my enemies to be "uprooted, smashed, destroyed, lowered and subjugated"? Maybe I wish for them understanding and wisdom to realize that they're doing wrong so that they can repent. Is it a problem that I feel this way? Can I maybe replace this part with my own prayer?
While you ask your question from what some may call a "modern moral standpoint," this sentiment is actually expressed in the Torah and is part of halachah (Torah law), as can be seen from the following episode in the Talmud:
There were certain hooligans who resided in the neighborhood of Rabbi Meir, and they caused him much misery and anguish. Once, Rabbi Meir prayed for mercy regarding them, so that they would die. His wife Beruriah said to him, "What makes you think that such a prayer is permitted? Is it because the verse states 'Let sinners [chataim] cease from the earth'? But is it written 'chotim'—sinners? Rather it is written 'chataim'—that which causes one to sin, namely the evil inclination. Furthermore, the end of the verse continues, '…and let the wicked be no more.' Since the sins will cease, there will be no more wicked men!
"Rather," she concluded, "pray for them that they should repent, and there will be no more wicked people."
He did pray for them, and they repented.
As we can see clearly from this episode, one should not pray for others to be punished, rather we should pray that they repent and do teshuvah. Which brings us to your question: How can we then do the exact opposite and pray for the wicked to be punished three times a day in an established prayer?
Furthermore, if we look at the book of Psalms, only a few chapters after the verse that Beruriah cited to teach us that one should pray for the cessation of sins rather than the punishment of the wicked, we find King David doing the exact opposite!
We find King David praying that "when he [King David's enemy] is judged, let him emerge guilty, and let his prayer be accounted as a sin. May his days be few, and may someone else take his office of dignity. May his sons be orphans and his wife a widow. May his sons wander, and [people] should ask and search from their ruins. May a creditor search out all he has, and may strangers despoil his labor. May he have none who extends kindness, and may no one be gracious to his orphans…" Hardly a manifestation of the above cited dictum, based on King David's own words, to pray for the cessation of sins but not sinners!
Back to the prayer we are discussing, which is one of the nineteen blessings of the most central Jewish prayer, the Amidah. Originally, the Amidah contained only eighteen blessings. In the generation that witnessed the destruction of Jerusalem and the Temple by the Romans, the Jews were further plagued by members of various sects who took delight in informing on the rabbis to the Roman authorities, and in devising various ways of enticing the unsuspecting Jews to abandon the path of their forefathers. It is against this backdrop that the Talmud relates:
Shimon the cotton merchant arranged the eighteen blessings of the Amidah before Raban Gamliel in Yavneh. Raban Gamliel said to the sages: "Is there anyone who knows how to formulate a blessing against the heretics and the wicked?" Shmuel Hakatan ("the small") arose and formulated it.
A little knowledge of the author of this prayer will help us to understand it better. As we know, every word in our prayer is exact, each word reflecting a deeper meaning and intent. It was for this reason that Raban Gamliel, who was the leader of the Jewish community at the time, sought a unique individual that was suited for the formulation of this unique blessing.
Examining Shmuel Hakatan's other sayings (there aren't that many), we find a character that appears in striking contrast to the Shmuel Hakatan who composed a prayer against the heretics. In Ethics of the Fathers we read:
Shmuel Hakatan would say: "When your enemy falls, do not rejoice; when he stumbles, let your heart not be gladdened. Lest G‑d see, and it will be displeasing in His eyes, and He will turn His wrath from him to you."
Shmuel Hakatan is simply quoting the Book of Proverbs, yet the Mishnah cites this in his name. Why? Because this was a statement that typified him; something that would be continually not just on his lips but reflected in his actions. Never mind not praying for the punishment of the wicked, Shmuel Hakatan is of the opinion that even when they are already punished we should not rejoice. Hardly the one you would expect to rise to the occasion of formulating a prayer for the wicked to be punished!
All this leads us to the conclusion that there must be different types of sinners. For some we are permitted to pray that they be punished, while for others we are not.
Again we quote the Ethics:
One who causes the community to be meritorious, no sin will come by his hand. One who causes the community to sin is not given the opportunity to repent.
Commentaries explain that this means to say that Heaven does not help such an individual repent. The doors of repentance are never locked, but for some people, they may require a lot of pounding before they open.
This is the sort of person to whom the blessing against the heretics in the Amidah refers. Yes, we would all prefer to pray that this person repent rather than be punished. Furthermore, even if we could not pray for him to repent for whatever reason, we could just leave him alone and out of our prayers altogether—as long as his actions are not affecting anyone else. Here is an instance, however, where we are simply left no choice. We are told that Heaven does not help him to repent, so of what use is it to pray for him? Yet neither do we have the option to leave him be while he causes others to sin, informing upon us to the authorities and endangering the entire nation physically and spiritually.
Imagine a patient who has a limb with a malignant infection that is slowly spreading throughout the body, so that his only hope for survival is to have the limb amputated. Any doctor who out of misplaced mercy for his patient decides not to amputate, not only is he not helping his patient, he may very well be condemning him to death. While no sane person with even an ounce of compassion wishes to see someone else suffer, let alone lose their limb, when faced with an existential threat we are many times left with no other choice.
Now that we know this, we can better understand the wording of the blessing. We say, "…and may You swiftly uproot, break, crush, and subdue the reign of wickedness speedily in our days." The order of the words is puzzling. If the reign of wickedness is uprooted, broken and crushed, how can it now be subdued? What is left to subdue?
The answer is that this wording follows a teaching of the Zohar, that there are four general forms of evil forces in the world. Three are completely lost to evil, while the fourth one, known as klipat nogah, is a composite of both good and bad. It turns out that in the blessing against the heretics, while we pray for the total eradication of unsalvageable evil, we also pray that whatever good there is mixed up in there should indeed be saved. As the Rebbe writes regarding this part of the blessing of the heretics:
…pause slightly between the words crush and subdue, in consonance with the intent that uproot, break and crush refer to the three forms of evil that must be completely eradicated. Subdue refers to kelipat nogah that needs only to be subdued and can be purified.
May we merit the day when all evil ceases to exist with the coming of the ultimate Redemption.