An additional, somewhat oversimplified, way to look at the AGC control loop is this:
If you consider the transistor (or other type of amplifier in the control part of the loop) provides 180 degrees of phase shift then an addition delay of 180 degrees is all that is needed for the control loop to go from being a control loop to an oscillator. This can be done with as little as two equal RC time constant type delays chained together.
Assuming there are delays in the AGC control loop due to other components not being infinitely fast, or other circuit compromises, you can make sure one of the RC time constant delays is much larger than the rest. That will swamp out the effects of the other delays. Many AGC circuit go even further by causing the AGC to cut in fast but restore the gain at a slower rate.
If there was no delay in the control loop the AGC might be so good as to adjust the gain microsecond by microsecond and result in an output signal (audio) of zero volts AC, across every cycle of the audio.
Finally if you are controlling a system with a lot of inertia (such as the hot water temperature to your shower head, air conditioning, motors or the melting temperature of the heated head of your 3D printer) another type of control loop, called the PID loop, is used.
In the PID loop the rate of change of the temperature/speed/whatever is fed into the control voltage to help the loop anticipate when to shut off (or turn on) just ahead of the desired value, so the temperature/…/… will arrive at but not overshoot the desired value (set point).
The PID loop would likely not be as beneficial in an AGC circuit when your neighbor down the street keys up his kilowatt with his beam pointed your way, at there is nothing gradual to anticipate. So simpler control loops are the better choice for AGC.