Over two days, the Fed chairwoman spoke with five Republicans, some of whom she had never met privately since taking over at the central bank, according to her public calendar. She stuck to a script she had delivered many times, said a person familiar with the calls: The bill could allow politicians to interfere with Fed policy; academic studies show countries with independent central banks have lower inflation; the Fed is already audited.Ms. Yellen didn’t persuade them. Though the Senate voted not to move forward with the bill—a relief for the Fed—only one of the chamber’s 54 Republicans voted in the Fed’s favor...The person leading the institution isn’t a politician—she’s a macroeconomist who spent most of her career at the Fed and in academia. Yet the task ahead of her, now that Donald Trump is president, might require a different set of skills. The new president thrust Ms. Yellen and the Fed onto the national political stage by criticizing them sharply during the campaign, and his election raised expectations that GOP bills to rein in the central bank could become law.The president has also said he would probably find a replacement when Ms. Yellen’s term is up in February 2018, which means he would likely nominate a successor by late summer, rendering her a lame duck...Ideas include requiring the Fed to establish a mathematical formula to guide interest-rate policy, limiting its emergency-lending powers and forcing the central bank to return billions of dollars banks paid to be members of the Fed system. The phenomenon is apparent outside the U.S., too, with central banks from Japan to the U.K. grappling with skepticism of their efforts to boost their economies...The financial crisis, however, battered the Fed’s credibility. Many lawmakers and some economists want more information about how the central bank operates and what it may do in the future...Though Mr. Trump hasn’t said whether he would support the measures, his campaign remarks—such as accusing Ms. Yellen of keeping rates low to help Democrats—suggest he has no qualms about criticizing Fed policy or its leadership, a departure from the recent tradition of presidents staying mum on such issues... Former Chairman Alan Greenspan, who scheduled his own breakfasts with members of Congress, had extensive relationships in Washington when he became chairman and often operated as a one-man congressional-relations shop.His successor, Ben Bernanke came from academia, but developed a rapport with members on both sides of the aisle during the crisis. Those relationships later helped him beat back legislative efforts to strip the Fed of its powers to supervise banks... She has met two dozen times with members of Congress since November 2015, either hosting them for breakfast or lunch in a private dining room at the Fed or shuttling to meetings on Capitol Hill, in addition to logging more than a dozen phone calls...More typical is what happened when Congress considered tapping the Fed to help pay for federal highway programs. Ms. Yellen warned it could set a dangerous precedent. Congress took even more money from the Fed than initially proposed, including $19.3 billion from its capital account.