ROME—Mario Draghi, the former head of the European Central Bank, was sworn in as prime minister on Saturday to lead a unity government that has to steer Italy out of the coronavirus crisis and an economic slump. All but one of Italy’s major parties have rallied to his side and his cabinet includes lawmakers from across the political spectrum, as well as technocrats in key posts, including the finance ministry and a new green transition portfolio. Much now rests on Draghi’s shoulders. He is tasked with plotting Italy’s recovery from the pandemic and must immediately set to work on plans for how to spend more than 200 billion euros ($240 billion) in European Union funding aimed at rebuilding the recession-bound economy. If he prevails, Draghi will likely bolster the entire eurozone, which has long fretted over Italy’s perennial problems. Success would also prove to Rome’s skeptical northern allies that by …