"Airplane mode" is rather unfortunately named. Phones originally added it as a way to quickly comply with FAA guidelines that no radios should be transmitting within the passenger compartments aboard flights. This means no cellular service, no wifi, so no Amber Alerts as well. But it's now been modified to allow you to manually turn on wifi to take advantage of in-flight wireless Internet service. Some airlines are toying with adding in-flight cellular service as well, so if that catches on, "airplane mode" will no longer have anything to do with airplanes.
The Amber Alerts are sent over a special cellular broadcast channel - kinda like SMS, but a different channel reserved specifically for emergency broadcasts (EBS, tornado...