Oh, I do believe that God allows suffering to exist because God couldn't create us perfect. You are the one who believes that God could have created us perfect but for some reason chose not to.
I think God allows suffering because He uses it to teach us good and evil and give us a greater understanding of the universe that we might choose in and of ourselves to be good beings. He could not create us to be perfect like Himself so through our free will, we must learn to choose good over evil to become more perfect. If God had the power to create us perfect, He wouldn't need to allow suffering in the world. I do believe He is capable of creating a world without suffering, but for us to grow and learn He has created a temporary world and allowed suffering to exist. I do believe the narrative of the Garden of Eden and the reason for the fall of Adam and Eve was to give the greater experience in a fallen world of good and evil. A means of salvation was provided to grant all who die to live again immortally through the resurrection. Those who are willing to choose good over evil of their own free will and choice are candidates for the Kingdom of Heaven and are on the path to perfection. Heaven itself is a world created by God without suffering. So your proof is wrong. I never ever said God didn't have the power to create a world without suffering. You imagined that.