"Why can't the radio hams use something more modern, like a phone, to talk to people that are far away, rather than having to rely on AM radio, that clearly doesn't work because of all the homeplugs everywhere?"
It's not just them. Consider shortwave radio listeners who are trying to pick up reflected radio transmissions from far away. They already have a hard enough time as it is because of the distance; any significant interference and they just lose the signal. And no, for many of them Internet Radio is not a substitute, either because they don't see the need for Internet or because the station transmitted doesn't have an Internet version.