Why is amplitude-modulated radio broadcast (AM) still active?
In a nutshell: One antenna will give you a usable radius of 100...1000 km, depending on the power used.

In Germany, for the example of my favored news station Deutschlandfunk, we used to have two long-wave AM stations (153 and 207 kHz, IIRC), and I do miss them every once in a while. The one at 207 kHz covered pretty much all of Southern Germany, and while I admit that the quality was low (as in: landline telephone-ish low), you could listen to the program with no trouble, anywhere in your house, and understand every word well.

Now, for terrestrial distribution, they just use FM, which works in a few small places only, or you could try DAB+, and I'm not sure if the latter works in all places. I do miss the robustness and the beautiful simplicity of long or medium wave AM.

It's not so much the type of modulation (AM vs. FM). It's the low-ish frequencies that tend to work well over wide areas and even through big walls, for example if you're downstairs.

It's not true that no one ever listened, and in contrast to North America, for example, Germany used to have only very few good stations on AM in the decade before they pulled the plug on it, which gives you another very important reason why few people listened.

A personal note: It twists my stomach to see how AM has already vanished, and to know that some want to abandon analog FM as well.

If you were to get cynical, you could argue there is some strong political will to seriously srew up anything terrestrial for good, at least in Germany. A bit off-topic here, and a rant, but terrestrial TV broadcasting shows you how bad it can become, and it's a fine example of unclever engineering: Analog terrestrial TV was shut down not long ago, in the early 2000s, with DVB-T as a replacement. Soon, (mostly private) stations stopped broadcasting on DVB-T, and now, DVB-T2 is about to be introduced, and of course, it's not backwards compatible to DVB-T, so any DVB-T receiver will be a piece of useless junkTM very soon. Considering the beauty of analog TV, this is sickening. There was black-and-white TV. Then they figured out how to put color into the signal while black-and-white receivers would still decode black-and-white and the new color TV signal, and color TV receivers would decode old black-and-white signals just as well as new color TV signals. Then, they put all other sorts of fancy stuff into the signel (stereo, videotext, ...) and everything was still forward and backward compatible. That's what I call good engineering, even more so if you put it into the context of its time and consider how advanced things were with respect to what was possible with the available technology.