It's a great paradox but not one you can use to disprove the existence of God. Or, I should say, you can use it to disprove your idea about God, but not God as He is described in the Bible.
Let's take an example from the Old Testament (Numbers 21:5-9).
And the people spoke against God and against Moses, “Why have you brought us up out of Egypt to die in the wilderness? For there is no food and no water, and we loathe this worthless food.” Then the LORD sent fiery serpents among the people, and they bit the people, so that many sons of Israel died. And the people came to Moses, and said, “We have sinned, for we have spoken against the LORD and against you; pray to the LORD, that he take away the serpents from us.” So Moses prayed for the people. And the LORD said to Moses, “Make a fiery serpent, and set it up as a sign; and every one who is bitten, when he sees it, shall live.” So Moses made a bronze serpent, and set it up as a sign; and if a serpent bit any man, he would look at the bronze serpent and live.Deadly snakes came upon the Israelites because of their sins. The people asked for relief from the snakes and they got it, but not in the form they wanted. Note that God didn't kill or drive out the snakes. Instead he gave an antidote: a bronze serpent that the Israelites were to look upon.
What kind of antidote is this? A statue of a serpent cures snakebites? In the Gospels, Jesus uses this passage to describe His own death (see John 3:14). The Great Commentary of Cornelius a Lapide summarizes the connections succinctly:
But perhaps there's a more immediate lesson to be learned here. God, in His goodness, is slow to destroy. Instead He gives us a cure which we are free to choose. So He does not destroy evil. To do so would rob us of our free will. Instead He places, alongside evil, a greater good that we may choose.
Most fully does S. Chrysostom draw out the analogies between the brazen serpent and Christ. He says, “Lest any one should say, 'How are those who believe in the Crucified One able to be saved, when he did not deliver Himself from death?'” He brings forward the ancient history. For if the Jews by looking at the image of a brazen serpent were freed from death, how much greater benefit will they enjoy who look to the Crucified Redeemer? For by the one the Jews escaped temporal death: by the other believers escape everlasting death. There the suspended serpent healed the wounds which the serpents had made: here Jesus, nailed to the cross, healed the wounds inflicted by the incorporeal serpent (the devil). There those who looked with their bodily eyes obtained the healing of the body: here those who look with their spiritual eyes obtain the remission of all their sins. There a serpent bit, and a serpent healed: here death destroyed, and death hath saved. In the one case the serpent which destroyed was full of poison, and delivered no one from poison. And in the other case the death which destroyeth had sin, as the serpent had poison: but the Lord's death was free from all sin, just as the brazen serpent had no poison. You see how the figure answers to the reality.
Secondly, actions have consequences. In the story of the seraph serpents, the people grumbled against God, accusing Him of bringing them into the desert in order to destroy them. Note the distinction: the Israelites were hungry in the desert. Moses, acting on God's behalf, had brought them into the desert. It seems reasonable to ask Moses (and God) for food and water. But rather than ask for food the Israelites accuse God of murderous intent, of an ulterior motive which had now been revealed. That deserves some response from God.
But just as the sufferings were brought upon by their disobedience, their salvation was made possible by their obedience. In this case, the act of looking upon the serpent. It seems an arbitrary and almost nonsensical demand: just look at the serpent. Yet it's an easy way for them to demonstrate their obedience to God. And God never makes impossible demands.
Pope Emeritus Benedict XVI acted this way. When he was elected Pope, most people expected him to swiftly and aggressively crack down on dissent and flaky liturgies. While Pope, he did crack down on some goofball bishops and theologians, but he primarily taught the truth. He didn't wholesale excommunicate Theology departments at so-called "Catholic" universities, but instead promoted orthodox teaching: even to the point of writing his own thoughts down. Likewise he didn't purge the church of tie-dyed liturgists, but instead promoted reverent and beautiful liturgies, starting with his own Masses but also in the promotion of the Extraordinary Form and Anglican liturgies.
It may be disappointing the God or the Church doesn't smite the evildoers as often as we'd like. Criminals run free, banal liturgies are the norm, dodgy theology is taught in schools and universities. But God and the Church have patience. At some point WE might be the fiery serpents to someone else. And when we are in that position, we'll be happy that God doesn't just send a lightening bolt, which we might deserve, but instead gives us a chance to repent and reform.
