Why does a phone's flight mode also disable FM radio?
All broadcast-band radio receivers contain local oscillators which are used in their intermediate frequency (IF) circuits to downconvert the (higher) frequencies they receive. These IF circuits are prohibited by law from radiating more than very small amounts of radio frequency power while operating, but to prevent any possible chance of having the airplane's navigation receivers pick up interference from portable radios, they have to be shut off during flight.
This is a minor problem today, but old-school shipboard radio receivers during WWII were supposedly capable of radiating enough power off their IF stages that submarines could tune in on them and use them as homing beacons to track convoys and then sink them.