Why are charge pumps only used for low current applications?

There are two problems with your idea. One practical, and one fundamental.

The practical problem is that per amount of stored energy capacitors are more expensive than inductors, and on top of that the realy high-capacity capacitors (electrolytic) age.

The fundamental problem is that charging a capacitor from a voltage source is fundamentally lossy (you dissipate heat). This might seem counter-intuitive, but is nonetheless true. (There was a question about this some time ago.) Hence a flying-capacitor voltage converter, even an ideal one, is inherently inefficient. (An ideal inductor-based voltage converter is 100% efficient.)

You might think it strange that the world is unfair to capacitors, but that is our human fault: we supply power mostly from voltage sources. For current sources the inverse is true: an ideal current converter from flying capacitors can be 100% efficient, while one from inductors must necessarily be lossy.

Capacitors would be better if the source and output were constant current. You could charge the capacitor until the voltage rose to a certain level, then discharge the capacitor into the load impedance to maintain a constant output current. You'd use a big inductor as an output filter to maintain the output current constant.

Since our sources are constant voltage and we usually want constant output voltage, using inductors to store energy and capacitors to filter it makes more sense.

Note that all efficient switching supplies have both capacitors and inductors.

Yes, charge pumps (flying capacitor) can take a voltage and move it around, flip it, even multiply by integers and such like, but every time you charge or discharge a capacitor through a resistive switch you lose a portion of the capacitor's energy change in the switch itself - a larger voltage change means more losses. A lower resistance switch just means that the energy lost for a given voltage change is compressed into a smaller slice of time, the total remains constant.

If two capacitors or series strings of capacitors with different voltages are connected together, their charges will average out in a way which reduces the amount of energy stored therein. If they are connected using an inductor, the excess energy will be transferred to that inductor and may subsequently be put to some useful purpose. If the connection is purely resistive, the energy will be 100% converted to heat. Minimizing the resistance will not reduce the energy loss; it will merely reduce the amount of time required for it to occur.

Consequently, in order for a charge pump to be efficient, the capacitors need to be large enough that voltage across them never varies very much. In cases where a charge pump doesn't need to convey much energy, one can use a linear regulator on the output and boost the voltage enough that under worst-case ripple conditions the output voltage will still be high enough to maintain regulation, but efficiency will be limited by the ratio of the load voltage times the boost ratio to the source voltage.