You already assume that this circuit has a direct application and that it is supposed to be an amplifier (you mention voltage gain) or buffer. Why would you assume that it has this function?
In the book this circuit is used to explain the basics of a switched capacitor circuit. There is no direct mention on how to use it.
In general such a circuit consisting of S1, C1 and S2 behaves as a resistor. Remove C1 and replace S1 and S2 by a single resistor. Now how would the charging curve for charging C2 look? Do you spot the similarity?
It behaves the same way as a simple first order RC lowpass filter. This is one application of this circuit: switched capacitor filters. This is used to filter / process signals which could carry information. In power electronics there is no signal (carrying information) but the circuit can still be used.
If you load the output and control the switching of S1 and S2 in a certain way that depends on the output voltage (in a feedback loop) you can use this as a voltage regulator. That is the point of the book, I believe, as it is about switched capacitor power converters.