Here are the basic reasons to have a large central bank for the US, and in fact, any modern economy functioning within a democratic environment:
Profit-motivated interests of the Banking Industry are subjugated to the directions of the Federal Reserve, which through its policies and regulations, protect the depositors and creditors of individual banks.
Interbank transactions are underpinned by the Federal Reserve, as it acts as the "Lender of Last Resort".
The Congress has established three main goals for the Federal Reserve: Maximum Employment, Stable Prices, and moderate long-term interest rates. The latter two are achieved first, and with the establishment of that environment, then a good employment picture substantially follows.
How does the Fed achieve stable prices and moderate long-term interest rates? This is done basically through two mechanisms which essentially "defend the currency" of the US. By exercising direct control over short term interest rates, and the overall money supply, the value of the dollar (usually) changes only very slightly from year to year, in terms of its purchasing power for US-made goods and services.
Additionally, the Fed defends the dollar through management of our own faith and confidence in our banks, with which most economically active people have a great deal of activity via mortgages, savings and checking accounts, consumer loans, etc. etc. Federal Reserve regulations establish guidelines and parameters within which banks must operate, and they monitor the adherence to these regulations to varying degrees. In such instances when the Fed has not been closely watching or enforcing adherence, many banks engage in more risky loan and investment practices, and the result is always some sort of general financial crisis or collapse, just as we recently experienced.
In its most vigilant state, the Fed literally has its hand on the throttle of USA's "economic train". If the throttle is set too low, the whole train grinds to a halt, or may start sliding backwards down the hill. If the throttle is allowed to run too fast, derailment is predictable certain. And, in times such as we find ourselves in now, the throttle is wide open, and they are shoveling fuel into the boiler as fast as they dare, by way of increasing the money supply monthly, as QE3 repurchases distressed bonds at a rate of $85B per month. This is all in an effort to maintain and gradually increase the forward momentum of the economy, and get us moving back uphill and forward again.
Without the Fed, none of this would be happening, and individual random market forces would be thrashing about, without overall direction, and only with self-interested or self-protective purposes. The result would be economic disaster beyond comprehension, with destroyed banking, markets and investment systems, unemployment at levels equal to or above that of the Great Depression, and no confidence in or hope for an economic recovery.