He gives them $600m so that he controls the board, how would that benefit him vs. just spending the money elsewhere? He doesn't get the Church's shares, just their vote.
You are mistaken, Michael DOES get the Church's shares in Immobiliare
"The Corleone’s are prepared to deposit 500 million dollars [later 600 million] in the Vatican bank at such time as Mr. Corleone receives majority control of Immobiliare."
Michael knows that the head of the Vatican Bank, Archbishop Gilday, has accumulated a massive deficit and offers $600 million in exchange for shares in Internazionale Immobiliare, an international real estate company, which would make him its largest single shareholder with six seats on the company's 13-member board. He makes a tender offer to buy the Vatican's 25% share in the company, which will give him a controlling interest. Immobiliare's board quickly approves the offer, pending ratification by the pope.
By doing so he becomes the largest shareholder and gains six of thirteen seats on the board of the largest real estate company in the world which is entirely legitimate and non-Mafia related, something Michael has been trying to achieve for many years/
DON ALTOBELLO Your old partners, are grateful for the money you’ve made them, they, they worship you. But their hearts are broken, because, they think you abandoned them. They want to share your deal on Immobiliare. To be a family again. It can purify their money. MICHAEL This I cannot do. Immobiliare, must be legitimate.