“Vision of the world’s desire” is an allusion to the novel The World’s Desire (1890) by H. Rider Haggard and Andrew Lang. In chapter 2, also titled “The Vision of the World’s Desire”, the goddess Athena shows Odysseus a vision of Helen of Troy:
‘Nay, Odysseus, didst thou not once give me one little hour? Fear not, for thou shalt not see me at this time, but lift thy head and look on The World’s Desire!’
Then the Wanderer lifted his head, and he saw, as it were in a picture or in a mirror of bronze, the vision of a girl. She was more than mortal tall, and though still in the first flower of youth, and almost a child in years, she seemed fair as a goddess, and so beautiful that Aphrodite herself may perchance have envied this loveliness. She was slim and gracious as a young shoot of a palm tree, and her eyes were fearless and innocent as a child’s. On her head she bore a shining urn of bronze, as if she were bringing water from the wells, and behind her was the foliage of a plane tree. Then the Wanderer knew her, and saw her once again as he had seen her, when in his boyhood he had journeyed to the Court of her father, King Tyndareus. For, as he entered Sparta, and came down the hill Taygetus, and as his chariot wheels flashed through the ford of Eurotas, he had met her there on her way from the river. There, in his youth, his eyes had gazed on the loveliness of Helen, and his heart had been filled with the desire of the fairest of women, and like all the princes of Achaia he had sought her hand in marriage. But Helen was given to another man, to Menelaus, Atreus’ son, of an evil house, that the knees of many might be loosened in death, and that there might be a song in the ears of men in after time.
H. Rider Haggard and Andrew Lang (1890). The World’s Desire, pp. 18–19. London: Longmans.
In the Greek myths, Helen is the most beautiful of women, desired by all, hence “the world’s desire”. In “The Oracle of the Dog”, the sudden apprehension of how he can clear his debts and restore his fortunes by secretly murdering his uncle strikes the young man with as much force as the sight of Helen of Troy struck the princes of Achaia. It is quite a grotesque comparison: monetary greed likened to sexual cupidity.
“Too respectable” is a paradox—of course murder is not respectable. There are two ways to resolve it, depending on how we interpret “the devil himself would hardly have induced”. If we take this to mean “the devil himself would hardly have wanted to induce”, then what Father Brown means is that killing one’s rich relation in order to inherit their fortune is a commonplace, ordinary kind of murder, the kind carried out by ordinary and respectable people, and the devil is not interested in that kind of murder. But if we take it to mean “the devil himself would hardly have been able to induce”, then what Father Brown means is that the young man’s self-conception would not permit him to carry out a murder in a commonplace manner, but that a daring and risky chance was something he could not resist.