BTW, on closer examination of the circuit, it appears to me that the
LM555 is switching a relay which in turn switches rectified but
unfiltered mains voltage to the oscillator. I suspect that the piezo
element determines the frequency of oscillation, and this oscillation
appears to be amplitude modulated by the 120Hz/100Hz mains.

Yes, the 555 is switching the relay (on/off).

I don't know what exactly this design is, but it seems you are right. The
transformer has three pins. Two of them drive the transducer, and the other
one is connected to the mains power. Parallel to the driving coil there is a
capacitor.
In the first I thought a simpler design. Now, I think the schematic is
needed.