On a recent afternoon in the Senate majority leader’s office, Harry Reid decided he was done talking about his chief of staff, David Krone. Instead, he and his top aide wanted to chat about the Vatican. “We sure do like this new pope,” Mr. Reid said of Pope Francis. After a brief discussion about the pope’s opponents in the Catholic hierarchy, Mr. Reid, a Mormon from Searchlight, Nev., brought up the name of perhaps the pontiff’s most vocal critic, the American cardinal Raymond L. Burke, a once enormously powerful prelate who has lost key Vatican positions under Francis. “How about...