(Though not 100% sure why the radio station had power and the cell towers didn't
There are hundreds, if not thousands, of cell sites per provider in the same geographical area that an FM radio station covers. Thousands of generators that need refueling for cellular, versus two for FM (the studio and the transmitter). It's much easier, logistically, to keep FM up and running during disaster than the cellular network.
An authority is a person who can tell you more about something than you really care to know.