Q2 is the "power switch".
Both transistors are required for, and part of, the oscillation feedback loop.
It's based on an ancient and really vile (to put it politely!) relaxation oscillator circuit that has been around since the 70s or before, which both shorts the input power, and puts ludicrous current through the transistors each time it triggers (if the power source is capable of any significant current).
It's an inversion (positive<>negative & PNP<>NPN swapped) of this type of old "siren" circuit, which I first saw about 50 years ago; it was equally vile then!
(There should ideally be stickied warnings about circuits like this in every technical forum!)
Edit: The original (patented version), having just found that, put the load in series with the power supply, rather than just one transistor - so the "short circuit" current was usefully used, rather than just flattening the battery and heating the transistors for no purpose.
In the "boost converter", Moving Q1 emitter to the L1-D2 junction would have the same effect & should make the circuit far more efficient.