Is it possible to build an oscillator from an NPN transistor?
A transistor is saturated enough by this voltage to pull the cap to ground and discharge it

The problem with that is that an NPN transistor does not switch on/off abruptly. If it did your solution would work. There are Unijunction transistors which do just this.

But with an NPN it will discharge the capacitor to such a voltage where there's a balance, it will discharge it slowly such that the transistor's input voltage is such that it will discharge at the rate (slowly) I just mentioned. So: no oscillation! It will just sit there, consume current and nothing else.

It is possible to make a one transistor NPN oscillator if you add a transformer, the well known "Joule thief" circuit (for making an LED light up on a single 1.5 V battery) is an example of this:



Note how there is a (small) transformer there. It takes care of the phase inversion needed to make this oscillate. This circuit could be adapted to drive a piezo element instead of an LED.

Design challenges will be:


making this oscillate at an audible frequency (as is it works at around 100 kHz)
The LED limits the Vce of the NPN, a piezo element will not limit the voltage so maybe a zener diode is needed there.
getting the transformer right will require some trial and error.