When you provide a DC input, the circuit will provide ordinary voltage gain. (As you say, the input must be high enough to turn on Q1.)
This can be done with fewer components, however.
Then I wondered about the label 'tune' at the input. This suggests the input is supposed to be AC.
Could this be a frequency doubler?
I did some experimenting. I discovered it does work as a frequency doubler.
It requires (a) some amount of input resistance, (b) a resistor to load the output capacitor, and (c) careful adjustment of the bias current.
I would not be surprised if this was the schematic's intended function.