True, the device shown is a piezo element. These elements look like an entirely capacitive high impedence load to the driver circuit. I don't believe installing a pot in the circuit makes sense.
It might work. It might introduce some distortions that you can convince yourself are a change in volume. In my research in controlling volume of piezo transducers, I didn't see any reference to this technique. Yes, my starting point was an active beeper, but my solution could have been a piezo element, had I found a simple way to control volume - I did not. The only methods I found to control volume on a piezo element is to control the duty cycle of the driving square wave, which resulted in a modest volume change. I did quite a bit of research on this, and nowhere was a pot used to control volume of a piezo element.
Perhaps I am a crappy researcher and perhaps I know nothing above piezo transducers (only having used them for 30 years or so doing vibration analysis). My concerns here are somebody who thinks this is a piece of cake (but has never done this) who presents it as a fact to someone who has little knowledge of the subject.
Sorry, but you're making a big fuss over absolutely nothing, the volume from a piezo speaker is completely dependent on the voltage applied to it (exactly as a normal speaker), lowering the voltage lowers the volume exactly as normal.
The reason you're confused is that you were using an active beeper, where the electronics are built-in - thus you have no control over anything, other than the power supply feeding it.
It's a simple enough process, a piezo speaker (or any piezo element) bends in direct relation to the voltage applied to it, applying an AC voltage makes it bend back and forth, this produces audio. The higher the voltage the more it bends, and the higher the volume - apply too much and it cracks and is destroyed.
Have you never seen a crystal earpiece?, that's just a smaller piezo disk in a convenient plastic moulding - plug it in your radio and the volume control of the radio adjusts the volume in the earpiece perfectly.
Incidentally - a number of years ago now the UK magazine EPE published a freezer alarm using a PIC 12C508, a thermistor, and a piezo sounder. I had the capability of programming the OTP 12C508 series, so I built one as a matter of interest. I found the alarm wasn't really loud enough from 5 volts, so I modified the code and circuit (simply wiring the piezo across two I/O pins) to drive the piezo in bridge mode, This greatly increased the volume, as volume from a piezo is proportional to drive voltage, and my changes were published in the letters page of the magazine.