From a purely logical point of view, Christianity's free will answer is unassailable: it is only because we have the capacity for evil that good is possible, and without sin there would be no Redemption. QED.
But as you point out the comparative facility of New Yorker writers to make the existence-of-evil argument, it is also grating for the equally well-off to make the free-will counter argument. I have led a pretty sheltered existence, and I've never known true evil, in my bones or in my gut. I have not known real hunger, or real pain. The oh so neat argument of free will seems so cold, so utterly irrelevant, when speaking with, say, a Holocaust survivor who has given up on God after experiencing the camps ... Of course you can point out that it is in that horror that other Holocaust survivors have found a reason to believe in God, but such considerations seem equally useless when talking not about the general presence of evil in the world, but about the precise and unique evils that one person has suffered.
The best Christian answer to the existence-of-evil argument seems to me to be, therefore, not the existence of free will (although, again, it is a perfectly valid response), but the much more concrete reality of Incarnation. God allowed evil to exist but He loves Man so much that He defeated it not just through the abstract (yet essential) gift of free will, but also by embracing His creature's condition and experiencing evil in the same ways.
Of course, from an atheist's perspective, this begs the question: to believe that God mitigated the presence of evil by experiencing and defeating it personally is to believe that God exists. But there is another way to put it: if it were possible to believe simultaneously in the existence of evil and in the existence of a benevolent God, then this benevolent God would have to be the kind of God who is willing to suffer evil alongside His creature and with the same intensity. This seems to me to be a very compelling answer.
The first point - that nobody wants to hear about how the existence of free will requires suffering from someone who hasn't suffered meaningfully themselves - is part of what interests me about the correlation between material comfort and complaints against God for permitting evil to flourish, the rain to fall on the just and unjust alike, and so forth. I've always thought that how you respond to the Christian argument about free will and the necessity of evil depends in large part on how you respond intuitively to the experience of human existence - whether you instinctively regard life, the universe and everything as a Very Good Thing with a certain amount of evil and corruption woven in, or whether you regard human life chiefly as "a business of evenly rationed suffering," as Wood puts it, with the constant possibility that the "truly unbearable" will suddenly consume everything that you hold dear. (Wood employs the evocative term "hellmouth" to describe this all-too-common worst-case scenario, a term he borrows from Norman Rush, who defined it as “the opening up of the mouth of hell right in front of you, without warning, through no fault of your own.”)
Like the commenter quoted above, I've never experienced anything that fits the "hellmouth" description - and this in turn makes me suspect that my own essentially positive response to the universe, and my accompanying assent to the logic of the Christian position concerning its design, is shaped in no small part by my sheltered and relatively comfortable existence. Which is why it's intuitively odd to me (though I can see all sorts of reasons why it might be so) to observe that in many cases the correlation seems to run the other way, and that my experience - in which comfort breeds acceptance of God's purposes, rather than rage at His putative injustice - is quite often the exception rather than the rule.
As to the broader point - that the Incarnation provides a better answer to the problem of evil than the argument from free will, even if the latter argument is valid - I'm basically in agreement, though I think it's important to note that the Incarnation isn't actually an "answer" in the way such things are typically understood. To say, in response to someone who asks "Why does God permit bad things to happen?," that "God permits suffering because He wants us to be free, and true freedom requires the possibility of sin and death," is to attempt a logical explanation of the phenomenon of suffering (and to invite further debate on the subject, at least up to a point). To respond to the same question, on the other hand, by saying that "God so loved the suffering world that He humbled himself to become Man and share in our suffering, even unto death on a cross," is something of a argument-stopper, since it addresses the problem only indirectly. Try as you might, you can't making God's willingess to "suffer evil alongside His creatures" into an explanation for why He allows evil to to exist in the first place. In effect, you're answering a question by presenting the questioner with a mystery - one whose relationship to the original problem can be intuited but not proved.
That being said, I suspect that more would-be believers troubled by the existence of evil are persuaded by the mystery of the Incarnation than by any more direct answer Christian theologians have mustered - which should tell us something about the nature of man, and perhaps of of God as well.
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