What happens if the frequency of the receiver osc does not match (exactly) the carrier frequency?
With both AM and FM, there are sidebands on both sides of the carrier.

With AM there is a simple relationship between the sideband offset frequency and the modulation frequency but with FM the relationship is less straightforward. With both types of modulation the edges of the signal spectrum with be attenuated by the IF filter if there is a frequency error.

AM usually uses envelope demodulation and is quite tolerant of frequency errors. The pitch of the demodulated signal comes from the difference between the carrier and sideband frequencies, which remains constant as the error increases, so no pitch change. There is only moderate distortion with just the carrier and one sideband with AM but as soon as the carrier is filtered out, the envelope is destroyed and there is gross distortion.

Try listening to a single sideband signal with an envelope detector. With FM there will be distortion if some parts of the signal are lost. You can experiment with a broadcast band receiver but bear in mind that the filters are usually not very sharp.