The solution was to physically separate the power connections for the audio components. While @Dave mentioned in the comments something about a star ground topology, the problem was that while the ground connections were all centralized, the Vcc connections were also, and shouldn't have been.
In the original design, power was supplied to the entire assembly via a small PCB with header pins for each separate component. Two rows of male header pins, one for 12V DC and one for ground. In this arrangement, all of the noise was being dumped onto the audio components each time the servo motor was actuated. Adding capacitors helped slightly, but certainly not enough.
The main two wires supplying the small power distribution PCB are about 10 cm long. I added a second pair, and created a second power distribution PCB. The original one was used for the servos and receiver, while the new one was used for audio and amplifier. The "Y" split in the power cable ultimately had the effect of keeping the power clean on the audio side.
I also removed the ground connection between the RC receiver and Arduino, though this was unnecessary. Its presence did not add noise, nor did its absence cause triggering problems.
An electrical engineer coworker advised me that it was important to have separate power supplies for motors versus digital components. While this project still only uses one power supply, the solution was to split the power as early as practical, and dedicate each to its respective purpose.
@berto's answer and the comments on it refer to feedback, which normally results in a loud squeel which will continue until the amplfier gain is reduced.
Your problem appears to be something else. A possible cause could be a faulty automatic gain control or compressor/limiter in the amplifier.
Best Answer
We can only guess, but it is most likely a buffer, both to speed it up and to reference it to the 3.3V supply going into L1.
If I were implementing something like that, I'd make sure that the 3.3V supply was exceptionally clean, possibly even regulated from some higher voltage for no other purpose than to provide power for that AND gate. The reason for this is that the AND gate is essentially multiplying its supply voltage by the PWM -- that's why I'd make it clean, and that's why the circuit designer has put L1 and C27 in there.