> I honestly can't think of a better way of informing people of urgent events (natural disasters, mainly)

I know that recently, at least in my area (Flint, MI), they started pushing tornado warnings (tornado touchdown confirmed) to smartphones in a way similar to the Amber Alert OP discusses. It was actually pretty cool the night in June, when we had 5 tornadoes touch down in the area, since I don't have TV and was too busy to notice the weather outside to check the weather online.