In fact, if the authorities are not already doing it, they are failing in their duty.
If this were a criminal investigation, then possibly, but it is not.
It is a search and rescue and will remain that until the aircraft is found or information is received that suggests a criminal act. People on the internet speculating about a plane load of gold being hijacked by terrorists and taken to some remote island is not evidence that a crime has been committed.
Were the homes of the flight crew of AF447 searched while the plane was still missing?