There is another weakness. After all the well-planned murders to prove his point, he involves for the last two people, who recently moved in the city: Mills and his wife. Doe couldn't possibly have known about this.
I have two theories explaining this. First: Doe got under pressure after the police caught his trace. He had to switch plans, so the last killings were planned under pressure and not perfect. Second theory: He always planned the wrath involving the policeman going after him. He planned that after the first murders some detective will catch his trace. But to enrage him, he needed to hurt him really bad. So he planned to use the wife of whatever detective it will be. To match the scheme it had to be the Envy-murder, but as it had to be improvised. So it lacks the perfection he tries to achieve.