One way to answer this provocative question is to see this within the crisis of faith that permeates Modernism. Institutions such as church and state were deeply compromised in this era and the strong claims for truth and justice they espoused seen as empty. Certainly for Yeats, the Irish Catholic Church—but also Anglicanism—held less force, despite Yeats being a spiritual thinker. Eliot, too, changed religions, becoming Catholic in part because of its tradition of rationalism and deep traditions. Nationalism and colonialism were creating obvious problems of credibility. War in Europe and elsewhere exposed the failure of states and the propaganda that partially maintained their legitimacy.
For Yeats, then, the best people can to see these flaws of intellect and of ethics and of courage and to deeply suspect strong arguments for any particular view. Like Prufrock, the best of this Modern period seemed to lack a sense of authority untainted by the errors that had created the crises surrounding them. As Prufrock claims, "I am not Prince Hamlet." He seeks to abandon his agency, preferring instead to be an "attendant lord." Similarly, the gunman is perfectly balanced in the air, in and his values. He joined the air force because he wanted to fight but does not particularly care for the cause he defends or hate the cause he is meant to destroy. He achieves great self-awareness in Yeats's poem, but this awareness leads him only to an understanding that his death has no meaning and that his life has not been directed toward anything but a "waste of breath"—certainly not a conviction of value.
On the other extreme are the worst people, full of the erroneous convictions, unwilling to think deeply about the implications of the views they advocate. This passionate intensity, the violence that overwhelms thought, leads the airman to battle, causes the center to not hold, and devastates those who seek an alternative way through a crisis of thought and faith. In both "Prufrock" and "The Airman," we see the challenge to be heroic overwhelmed by the self-consciousness of the act, springing largely from a keen sense that one is not filled with the certainty needed to address the larger issues of the moment.
The historical context of these poems is important. The Yeats poems appeared in 1919, at the end of the First World War. "Prufrock" was published in 1915, at the height of the war. All these poems reflect a sense of alienation and political wariness. In "The Second Coming," the lines about "the best" and "worst" people can be understood as reflections on the war, or on the nature of revolutionary politics. The "passionate intensity" of the "worst" suggests that enthusiasm for political action can lead to a kind of mob mentality or a thirst for violence; the "lack of conviction" of the "best," on the other hand, suggests that years of conflict have made them disaffected and passive; they are unable to stand up for the kind of political change that is necessary.
This alienation appears again in "An Irishman Foresees His Death." In that poem, the speaker explicitly says that his will to fight did not come from any passion or love; he "does not love" the ones he guards, nor does he "hate" those he fights. In fact, the "end" he fights for he knows will not make much difference to "Kiltartan's poor"; he is lead only by a "lonely impulse of delight" in struggle.
"Prufrock" also deals with the moral alienation bred by the dehumanizing effects of the First World War. The speaker in "Prufrock" is indecisive and torn by his indecision; his thoughts, as represented in the poem, can be thought of in a way as those of one who has, in Yeats' words, "lost all conviction."
In both "The Love Song of J. Alfred Prufrock" and "An Irish Airman Forsees his Death," the speakers are alienated and detached from their own culture. The Irish airman says he does not "love" the people he is fighting to defend nor does he "hate" his enemy. He didn't enlist out of any great faith in the values of his country or community but simply out of a "lonely impulse of delight." He perceives his life primarily as consisting of "waste" and therefore doesn't mind if he is killed.
Similarly, Prufrock is alienated from his culture. He feels paralyzed, like an insect pinned to a wall, unable to speak his truth. The party he attends is dull to him, simply a repetition of many, many other parties he has attended, and he wonders if he has wasted his life in such meaningless activities. He longs to be part of what he imagines as the more vital and creative world of the ancient Greeks.
If the airman and Prufrock represent the best of men, or at least those of decency, we can see that they, in Yeats' words, "lack all conviction." The modern world makes it difficult for people to have the unthinking heroism of those in the ancient world because modern people are too individualist, too prone to see, critique, and be paralyzed by the faults they see in their own culture. They know the modern world is deeply flawed, and therefore have problems committing to it with passion and intensity. In many ways, what could be positive traits, introspection and self examination, have turned into an unhealthy detachment from life.
When answering this question, keep in mind the context of the poem in question: "The Second Coming" was written in 1919, amid a time of deep disillusionment in Europe. So, when thinking about the meaning of the lines in question, I'd suggest you think of it in terms of the very real soul-searching that characterized this period in history. How might you expect individuals to view their political leaders, as well as the future, especially given the experience of the First World War? What kind of vision does "The Second Coming" present, as far as it relates to those same kinds of questions? I think the lines in question are a very powerful illustration of the sense of disillusionment and cynicism that awakened in the time period in question.
From here, it's a matter of applying this same kind of thinking to the other two poems. When looking at "An Irish Airman Foresees His Death," it's worth asking: just what is the man's attitude toward the war as far as it relates to the enemy? to his own side? to himself? Similarly, consider the psychology of Prufrock. How does Eliot depict the world, as well as the character of Prufrock? Do we see the same modernist impulses at play? In thinking about these poems in these terms, I think you should be able to discern how they fit together in the manner your question asks for.