Disasters (natural or otherwise) are usually fairly localised. Cell towers have a reach of a couple km at best and can be easily overwhelmed, whereas AM radio can reach upwards of 100km at night from a single transmitter and doesn't care at all how many people are listening.

As you already mentioned, power is another issue, as a modern smartphone won't last past a few weeks, even when turning on for only 5 minutes once a day, whereas I've seen AM radios that can run continuously for at least that long.

Finally, my main concern: cell phones are complicated. An EMP (or flood) could fry all your electronics and you're basically screwed, but finding someone who can build a passable AM receiver (or even transmitter) out of various electronic junk isn't all that difficult.