Here is my take on piezo transducers. It is based solely on what I have read, so take it with a grain of salt.
I believe you can apply a DC voltage to a pz xducer and cause a DC deflection. It follows to me that you could apply a DC deflection and get a DC voltage, provided that you draw NO charge from it. That is the rub. Measuring the voltage would generally be done with an instrument with input resistance, i.e., 10Megohms., Using it to turn on a MOSFET, generally requires a shunt resistor to establish a grounnd reference, possibly many Megohms. Nontheless, that resistor will drain off charge, causing the output to be a differentiated version of the deflection.
I was thinking that, if you could predict the time when deflection was NOT applied, you could DC restore (e.g., clamp to ground) the voltage across the pz xducer (and gate), to remove stray charge. Then, when the device is not clamped, deflection could provide a voltage to the MOSFET gate, turning it on. There would still be time a limit, due to gate leakage current, but it could be a relatively long time.
I really don't know if a piezo can couple enough charge to the gate capacitance of a MOSFET to turn it on.