What I’ve never understood is how the FM receiver can lock on to the signal if its frequency is always changing. Doesn’t the receiver need to lock on to something? If the answer is “it locks on to the amplitude, which doesn’t change,” well AM is bad because the amplitude is subject to interference, so wouldn’t FM have the same problem?
One possibility is a phase-locked loop. I don't know if there's anything better. It matches the frequency of a voltage controlled oscillator to the frequency of the incoming signal by detecting the phase mismatch. Then, the control voltage for the VCO becomes the audio signal.
I don't think radios use a PLL to demodulate the FM audio: the signal has a huge "pilot" tone at 19kHz that you can match to get the first part of the spectrum, mono audio (L+R channels), and at double that frequency you know you'll find the stereo part (L-R channels). Precise phase estimation is only necessary to decode the RDS digital data (station name, datetime, etc.).
Most cheap FM receivers definitely use a PLL to detect FM. Giving automatic tuning is just a side benifit.