I guess it doesn't become any more "Breaking News" than localized government alerts. Sorry is this is in the wrong section, but a bunch of us were just in a meeting where we all got the same alert. We all have iPhones so we can choose to turn off alerts involving "imminent threats to safety or life" or "Amber alerts", but we cannot turn off any "Alerts issues by the President".
For those that are not familiar, here is some reference information: Wireless Emergency Alerts (WEA) | FCC.gov
Re: WEA Alerts
Someone will create a way to do that, though I am not sure why one would want to not know there was a national emergency like a 9/11 attack or something under way.