The zener diode circuit is fairly common, but in this case it is a bad design:
The capacitive dropper C2 acts as a current source. The 24V zener diodes limits the voltage to 24V generating 24V for driving the relay.
Right. Because the micro is only seeing the voltage difference directly across the zener, it only sees 5V. I still don't see how it doesn't clamp the main rail down as well. I thought to have a zener work properly it needs to pass a decent amount of current?
The 5.1V zener diodes generates +5V for powering a microcontroller or some other control logic. It has the series resistor R3.
The motor return is probably wrong, because it can't work the way you have drawn.
R6 could be used for sensing if the motor is turned on or maybe for sensing zero crossings.
Typically there is a series resistor for the capacitive dropper circuit, because if you turn the device on when the mains voltage is at peak of the sinewave, a very high inrush current flows for a short time. In your case C3 has probably a high ESR, therefore it couldn't handle the high peak current. So the zener diodes had to conduct the current and failed. I would check C3 if its ESR is <1ohms or just replace it.
I can't work on the coffee maker anymore. And I understand capactive dropper circuits, I just hadn't seen zeners used like this before.
I'm telling you -- the circuit is exactly as shown. I even have a picture of the backside of the PCB with the components annotated if you'd like to see it.
Here it is:

The motor connections are the very bottom left large pad, and the very large bottom right pad. You can see the only path for the current to flow is back through that same resistor (it's not going through the tiny wires over to the microcontroller board.)
Just want to say, this isn't my first reverse engineering rodeo, so to speak.