I found a much better place to connect the drain of the MOSFET. First I replaced R70 with a 510 ohm resistor that I happened to have in the same footprint.
Then I connected the MOSFET drain to the relay side of that resistor, i.e. to the point M2 on the schematic. This setup still shorts the incoming audio to ground and kills the pops, but leaves a voltage divider of R253 and R70 connecting the sidetone to the volume control. Depending on your value of R253, this may be perfect as is. My R253 was 220 k and I needed more volume, so I put a 50 k trimpot across R253. It is just about right for me at maximum resistance, but I can adjust it downward if I should want more volume still. If your R253 is a smaller value and the sidetone is too loud, just replace it with something of higher resistance.
The volume control will now have final control over the level and can be tweaked in operation if needed.