They're not in the emergency services business. Technically, neither is radio. When the feds created the Department of Homeland Security, they took on the emergency services responsibility. Radio stations are merely the conduit for that information.
That is a good point. As long as a station is on the air, the EAS system can "sieze" the signal to transmit local, regional and national alerts ranging from an approaching tornado system to an enemy attack. Even before Homeland Security existed, there was a structure involving local, state and national authorities who could initiate emergency broadcasts or bulletins even if there was nobody at a fully automated station. This has been expanded to include cellular devices. At some point, I expect there to be a system which creates emergency on-screen pop-up messages for other types of computers that use ISPs for connectivity.
An interesting angle to this is that the stations themselves can not initiate an EAS transmission, other than a test. The content has to come from designated authorities who can essentially "seize" the broadcast stations in the affected area and transmit a bulletin.
The police, Civil Defense and other government agencies essentially are on duty 24/7 waiting for bad stuff to happen. It's their job. And when it does, if the "bad stuff" affects broad areas, they can activate the EAS system.