"AM" is the modulation scheme. A more accurate name for this radio service is used in Europe, "middle-wave".

The modulation scheme is not the reason for propagating around mountains and bouncing off the ionosphere at night. The reason for that behaviour is the low frequency used. 540 Kilo-herz to ~1700 Kilo-herz. (I was told once by a young engineering colleague that 1,000 Khz was not RF. :-))) )

So the question is valid. FM could be used in the "middle-wave" band. It would reduce noise but might suffer more when bouncing off the ionosphere at night. (I have read this but never heard FM bounced)

The reason FM is not used in the middle-wave band is legacy regulation not technology.