> Plus, say a nuclear weapon went off anyways and all the FM transmitters died. Is anyone seriously thinking a lone AM transmitter 2000 miles away is going to help?

AM radio is simple enough to build with makeshift materials (https://en.wikipedia.org/wiki/Foxhole_radio). You don't need batteries or even a power source at all! (the radio waves themselves can bring power)

If civilization had a catastrophic event, I would guess that AM radios would be one of the few surviving technologies