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.
Narrow band FM is used for aircraft. police, fire, amateur radio communication and since 2021 it's even legal for CB radio in the USA. It's deviation is limited by law to about +/-2.5KHz, from centre frequency.
Setting up the modulation depth on an a broadcast FM exciter, I have used a spectrum analyzer and Bessel function information to make sure it was legal.
I've alluded to this elsewhere. As I mentioned, RF/RFI and spectrum management subjects have been downgraded or reduced in electrical engineering courses just about everywhere so it's little wonder we've problem.
FM by comparison is effectively line of sight, so with a practical broadcast antenna the upper limit is 40 or 50 miles.
And I should clarify… while it was certainly a cheap “system in a box”, it was a real multi component system with detachable speakers, etc. not a boombox or anything like that.
AM is more spectrally efficient than FM, and with synchronous AM receivers, you can get noise rejection that is near FM.