There are other emergency alerts. Amber alerts are for missing people. At least AFAIK but TBH I don't really care at this point it's a poorly though out and implemented system with shitty software that I end up disabling for better or worse.
It's unfortunate because the world would presumably benefit from a properly standardized and above all globalized way of subscribing to geographically local alerts of various sorts. My local government should be able to advertise their servers via the cell towers and I should be able to add and remove subscriptions from anywhere in the world as I see fit. And above all the messages should be properly authenticated. Last I checked the system was so half baked that it was trivially vulnerable to spoofing.