Single input controlled full H bridge for piezo buzzer
It is not advisable to leave DC across the piezo element, it can cause long-term unreliability especially in conjunction with humidity. 
The CMOS gates and buffers will have varying voltage drop depending on the current. They are as rail-to-rail as anything. Probably they are fine and may well have a lower voltage drop than a random discrete MOSFET if your drive voltage is insufficient for the latter. A discrete MOSFET may also have a lot of input charge, comparable to a small piezo element.
The piezo element does not have significant inductance, it looks (electrically) more like a voltage source in series with a capacitor, where the voltage source is a function of mechanical strain. If you whack the piezo there will be voltage generated at the terminals (this method is used create a spark to ignite the burners on some BBQ units, so the voltage is not necessarily small). It is recommended to put TVS diodes or similar protection across the piezo element to keep it from damaging the chip.