The problem is that overrunning clutch (it goes by different names) hidden somewhere in the back of almost everyone's bike.
If you pedal at a nice slow cadence and with decent pedal pressure, that clutch never overruns and 100% of your pedal effort is translated into increased speed. Not a bad way to ride. It works. This is how I ride.
Now, attempting to increase your cadence by gearing down brings you away from that 100% efficient comfort zone. It brings you closer to the point where your clutch disengages and you are simply pedaling for effect with zero pedal pressure and zero power transmitted to the wheels. Wasted energy.
Other common vehicles you may be familiar with don't feature an overrunning clutch. A manual transmission car, for example, locks the power to the wheels and functions essentially like a fixed gear bike. An automatic transmission car has a fluid coupling, so if the power is faster than the wheels, that fluid starts to drive vanes in a torque converter, and the car accelerates.
If you really want to develop a faster cadence, you have three options. You could simply accept that you will spin out the overrunning clutch from time to time and lose power. Second, you could increase your momentum (typically by going faster). This makes it easier to keep from spinning out because physics will encourage your bike to keep moving at the speed it is at. Finally, you could learn to sense the pressure on that overrunning clutch and pedal just slowly enough to keep it engaged.