This question has two parts: How can sin exist if God didn't construct its reality, and why did He do this? The first question doesn't involve God's omnipotence as an obstruction to the culpability of free will, but asks how God can create the outlines of the sin itself without being morally culpable. For example, a murderer can kill only because God allowed the technical existence of murder. The second question simply asks why God did this, and why not just let free will be unable to choose wrongful actions.
I'll start with the second question. The answer that we would become robots does not explain anything because it's neither true nor relevant. Free will is not eliminated when handicapped: I have free will whether I can fly like a bird or not, and I still have it even if the thought never occurred to me to fly like one. This is known as a Frankfurter, and famously refutes the idea that free will must have two or more options to be free: an assumption that actually predicates free will upon the environment and contradicts the very thing it presumes to support - its independent freedom. Moreover, it's hard to understand who exactly would be displeased if all hellbound people had been robots and never committed sins: all would be happy.
The answer has to lie in the fact that certain limitations, intuitively, prevent free will's true expression and thus deny an accurate assessment of culpability, making it unfair. As noted above, a Frankfurter example proves that free will can have only one option and still be free, since that could be the option chosen. However, for this to be true, it means that the other options, which by definition are denied, meaning they're nonexistent, must also be real. This leads to the conclusion that counterfactuals are real in the sense that they are genuine possibilities. This is supported by the idea in ethics that intent and not circumstance (energy in space-time) measures culpability. If intent is to be distinguished from space-time energy meaningfully, it cannot be a direct product of it - in other words, free will can't be predetermined, or more specifically preconfigured and preprogrammed outside of the free will-person. Therefore, the elimination of technical sin is not only unusable, it's logically impossible!
So how can God create a murder scene without being at least partially responsible? Is sinlessness simply logically impossible? But this conclusion confuses intent with definition. If I quote someone, I'm not agreeing with them. Sarcasm disagrees with words or actions by literally agreeing or acknowledging them. What the objection implicitly asks is how someone's free will can be guilty when God is omnipotent and created it, a question dealt with elsewhere. Like quarks which come into existence out of nowhere, something can exist by being uncreated.