Yes, I know about half- and full-wave rectification and power-supply filtering. Of course, the DC power supply circuitry has a filter cap that recharges every half cycle. I meant that the cap is not merely charged once to supply the chime's circuitry for the entire duration of the chime's song.

The DC power supply is energized 24/7 because of the diode, even though it's only used when someone pushed the button. It's the full-wave AC that is supplied to the chime only when the button is pressed. They need both constant power and a way to tell the button has been pressed.