There's been an official response now: The system was overwhelmed because instead of just one nation-wide alarm, several local institutions used the system for local alarms. Which is scary: The number of local alarms was three I read (very uncertain about that number). And the nationwide alarm came 31 minutes late. That seems unreasonable, but yeah I guess the test was productive at least.
Yes, obviously using the phone network is the better idea. But realistically the goal of these projects is not to create a working system. The goal is to create something "new" to leech money off the federal and EU projects, which they succesfully did.