One of the biggest problems with the Argument from Evil is that the conclusion of the argument doesn’t sit well with all of the various things that are presupposed in the argument itself. For instance, theists often argue that atheists can only make the argument by appealing to objective moral values. If atheists claim that God has to prevent suffering, then they’re saying that there are certain moral values that even God must obey. And if atheists claim that suffering is evil, they are presupposing some sort of standard that we use to distinguish between good and evil. They can’t say that it’s really all subjective. They can’t say that these moral values developed through evolution to preserve us as a species, because then these moral values would only apply to us, not to God. So the atheist has to appeal to objective moral values. The problem is that objective moral values point to a transcendent source of moral values, not to atheism.
This is a problem that theists usually point out, but as far as the Argument from Evil goes, atheists actually need a lot more than objective moral values to get the argument off the ground. For instance, they need suffering, but for suffering there has to be humans and animals (or something similar), and humans and animals are incredibly complex organisms. This complexity points to design, not to atheism. Atheists also need some sort of world where all this suffering is taking place, and any world will serve as the foundation for the Cosmological Argument. But there can’t be just any old world; the suffering we see around us requires a finely-tuned world. Otherwise humans and animals couldn’t survive. This, of course, is the idea behind the Argument from Fine-Tuning. Beyond this, atheists need minds to recognize the evil and formulate the argument, and this is part of the Argument from Consciousness. They also need a concept of God, because they’re claiming that this concept doesn’t apply to anything that actually exists. And our concept of God is used in various forms of the Ontological Argument.
The point here is that the Argument from Evil is an argument for atheism, and yet atheism doesn’t account for anything that’s presupposed in the argument. The argument requires objective moral values, extremely complex suffering beings, a world (a finely-tuned world, mind you), conscious minds, and a concept of God, and all of these are important elements in proofs for God’s existence.
If we consider this carefully, it turns out to be a tremendous problem for atheism. The thrust of the Argument from Evil is that theism doesn’t account for suffering, and that theism is therefore implausible. But let’s turn that reasoning around. Atheism, as far as I can tell, doesn’t account for anything that goes into its own argument. Atheism doesn’t account for the existence of the universe, or for life, or for objective moral values, or for anything else. Atheism accounts for absolutely nothing. So if we’re rejecting arguments based on their lack of explanatory power, we’d have to reject atheism long before we reject theism. In other words, it makes no sense at all to say, “Well, theism doesn’t account for evil very well, so let’s reject theism and believe in atheism, which doesn’t account for anything very well.” But atheists still make the claim.