The U.S. Congress established three key objectives for monetary policy in the Federal Reserve Act: Maximum employment, stable prices, and moderate long-term interest rates. Those objectives are as important today as they were a hundred years. The Fed also has responsibilities that go well beyond just monetary policy but also to insure the banking system remains sound. The role of the federal reserve has changed and expanded because the US and the world economy bears little resemblance to that of a hundred years ago. The US GDP in 1913 was 3% of what it is today. That economic growth over the last hundred years could not have occurred without the stability in the banking system created by the Federal Reserve and the tempering of erratic moves in the economy .
A discussion of abolishing the Federal Reserve is about as meaning as a discussion of abolishing money itself. You can't simply eliminate the functions of the fed. If you got rid of the central bank, you would need to push many of these functions to other regulators or private firms. For example, inflation has to be kept in check somehow. Prudential supervision is also important. The problem with eliminating the Fed is that you would need to delegate these responsibilities to another entity that could do them better and there is simply no such entity.