On multi-element piezos: many or most piezo pickups used to have individual elements, such as the Martin Thinline made by Fishman. The best live acoustic guitar I ever heard was a D28 with an old Thinline. Nevertheless, the thin ribbon pickups used today are an improvement over those old designs, in part because they don’t mess with the bridge as much.
I’ve tried lots of pickups and now have the K&K. It has three elements, so I’m probably still in the multi-element camp.
Electrically, the quack probably comes from the piezo’s similarity to a ceramic capacitor. With some exceptions, ceramic capacitors change their capacitance with voltage, so they distort sound waves. The higher the voltage, the more distortion, and even the typical guitar signal of 1 volt is enough. Furthermore, the distortion is the unmusical/harsh type - odd harmonics.
Most of the audible quack can be reduced with a simple circuit that disfavors those odd harmonics.
P.S. for the electronics nerds:
https://www.edn.com/signal-distortio...ic-capacitors/