I think a likely reason that AM radio is not included as an option on MP3 players is due to problems with static. Putting any type of motor near an AM receiver would likely cause static/noise when listening to the radio.

I've had the problem when listening to AM radio on a receiver while the TV is on. The noise is so loud that it is impossible to hear AM. FM is not affected.

FM is not susceptible to this problem and can be included as part of an MP3 player. It is possible that a player without moving parts (like a solid state player) would have the problem with static and could be included with a player.