The whole point of this, though, is integrating something new that works radically differently from existing aircraft. If other technological mechanisms can provide sufficient accuracy, why can't they be used?
It's not like they need to solely rely on AGPS either. Consumer IMUs have been advancing at a rapid pace (there's huge amounts of money being dumped in them due to gaming, VR, and mobile phones), and are capable of high accuracy when combined with an external reference. You can also use laser ranging, which is also very cheap these days (my robotic vacuum cleaner has a LIDAR turret on it, although the range would be less than the few hundred feet required here). If you know where you are, and you know the height above sea level at that location, and you know how far you are from the ground...
There are many tough problems to solve to make what Amazon is proposing practical, but accurately figuring out your altitude a few hundred feet from the ground is certainly not insoluble (or even particularly difficult). There are many things you can do to determine altitude at 300 feet than aren't possible at 30,000 feet.