Isn’t the “greater good” defense a logically fallacious response to the problem of evil?
It is logically fallacious. “The end does not justify the means.” (1759 CCC)
In the centuries theists have been using the “greater good” argument while unknowingly selling out the goodness of God. The atheists have been right in rejecting a “god” that ought to be rejected. The true God opposes evil with the fullness of his power. I believe the answer of evil is given here:
"Though omnipotent, God has irrevocably set a limit to his own power:
“God has not willed to reserve to himself all exercise of power. He entrusts to every creature the functions it is capable of performing, according to the capacities of its own nature.” (CCC, 1884)
In gratuitous generosity, God has arranged that real power in shaping the world be given to human beings and angels for the sake of imbuing created persons with authentic importance as co-creators. In offering himself totally to creatures, God has given himself away in a most radical manner: All powers and roles of importance that can logically possibly be entrusted to others have, in fact, been given to human beings and angels for the sake of imbuing authentic and irreplaceable importance to each creature made in the image of God:
“We can never give too great prominence to the Scholastic principle that God never does through Himself what may be achieved through created causality… For any result which does not require actually infinite power, God will sooner create a new spiritual being capable of producing that result than produce it Himself.” (Abbot Anscar Vonier, The Human Soul)
There are areas of responsibility that can only be acted upon by God (e.g. the creation of the universe out of nothing, or the governance of the entirety of reality via omniscient providence), and these cannot be given over to creatures due to the limits of logical possibility. Even so, the self-emptying of God is such that many of those actions which can only be accomplished by God himself (such as the forgiveness of sins, or the changing of bread and wine into the body and blood of Christ) have been entrusted to human beings as intermediaries through the sacramental ministry of the priesthood. According to God’s generosity, if something can possibly be done or mediated by a finite power, God creates a finite creature to do it rather than doing the thing directly. Since this is a true giving, and not merely the appearance of gift, it follows that creatures now have a kind of power in the world that God does not have.
Even though sin and the suffering caused by it is an infinite offense to him, God is (though metaphysically omnipotent) functionally dependent on the actions of creatures
obedient to him in order to manifest his will and justice in the world. God is in no way controlling things directly, and if a created person chooses to do evil, then real damage is done."