Consider that U.S. citizens do most of their shopping using checks rather than currency.?

The Fed controls the supply of money by controlling the interest rates. Higher interest rates means it costs more to borrow money so companies don't borrow. This reduces their supply of money. Lower interest rates means the companies are more willing to borrow and have the money to make products or expand factories. They don't limit your checking accounts. The Fed has to take some action. Taking no action is not an option.