The main critique of any post-axial religion is the Argument from Evil, but in the last few centuries we’ve added the critique that only atheism sees the world as it is, sc. as immense and without concern for some lone species of hairless primate. This might start as a critique of design but it is inseparable from the scientific disenchantment that moves out of the infancy of an anthropomorphism into the full grandeur of the tree of life, the majestically indifferent post-Copernican world, and Darwin fish.
These critiques conflict, perhaps fatally. The first insists that God has no answer to suffering and the second that the Christian world is infantile and therefore consoling. God cannot both save us from the indifference of the universe and have no answer to the suffering that we find in it.
I’m not appealing to dialectics here but more to experience. The sort of evils that someone might ground the AFE upon (the sudden or painful death of children) are the sorts of things that Christians overwhelmingly experience as throwing them into a pit so deep that God alone can be found at the end of it. The friends of those thrown “into the depths” feel compelled to speak and know they have nothing to say; compelled to be with them and know they can never be with them. Anyone might descend into Hell, but only Christ has gone there with any hope of returning. God is literally the only consolation one can experience in the depths of suffering, and so he is the only possible solution to the problem of evil. We can respond to this by saying it is all wish-fulfillment, infantilism, or pre-scientific enchanted magical thinking, but this commits us to declaring the AFE unsound.