It doesn't help that there are errors on the circuit diagram, L2 is 1.62mH (and L1 1mH ,as per text and parts list). L2 (1.62mH) resonates C15 (1nF) at 125kHz meaning that the amplifier has to drive a low impedance.
Since the drive to the amplifier before the filter /matching circuit (L1, C9 & C12) is a square-wave C15 will charge to half the 12v supply negating the need for split supplies.
With the push-pull drive the output impedance is low over the whole of the cycle, a single MOSFET would either be short or open for half the cycle depending on configuration (shunt or series feed), either would destroy the Q of the resonant circuit.
Note the Q of the resonant circuit increases the received voltage across C15 and hence sensitivity.
Jim