If I were going to erode public trust in emergency broadcast systems in order to increase the amount of damage I would do by launching an actual missile, this is how I would do it. A series of false alarms.

A similar, if obviously much smaller and less disastrous example, happens at my apartment about 4 times a week: the fire alarms for entire floors of my apartment building are easily triggered by people smoking in the breeze ways or burning their dinner. The result is that I routinely ignore fire alarms because the likelihood of a real fire is has been demonstrated to be exceedingly small.