This article accompanies the fable
The only writer I am aware of who played with this theme before the modern era is Hans Christian Andersen, whose dark fairytale The Storks was published in 1838. This story spread it around that babies could be found in distant marshes: “There lies in the pond a little dead baby who has dreamed itself to death,” said the mother (stork). “We will take it to the naughty boy, and he will cry because we have brought him a little dead brother..."
Today the white stork is the national bird of Germany, Belarus and Lithuania. Denmark and Finland opted for the somewhat similar white swan.
Above is a more traditional image from 1885, one of the last paintings by Carl Spitzweg: Der Klapperstorch. As delivery methods go, this one seems riskier than childbirth.