The story is a bit discombobulated in that it is not clear if a warning was issued and not received, or it was not issued in the first place.
This I do know, if you are not testing and doing practice EAS messages with your entire staff, when it comes time to use the system it will not get used. Out here in the West we don't have the tornadoes that cause many of the EAS warnings to be issued in other places in the nation.
In my 16 years of working at the state and local level of emergency management, I've only issued one EAS alert and that was for a levee that was being overtopped during a flood event in King County.
If you don't practice using the equipment or having draft messages composed, it is unlikely that someone will pull the trigger to issue an alert. In fact, they might not even consider doing it because they are consumed with other tasks.
Besides EAS there are other commercial systems that jurisdictions have invested in to call designated areas based on a polygon drawn on a map. But they cost money!
The best way to draw attention to you and your program is to have trouble with a warning. Let that be a warning to everyone else.