The article mentions that packing items in boxes is still done by hand, and I imagine loading/unloading trucks is still done with humans. However I can foresee completely-automated Amazon warehouses in the near future.
Amazon would need to have more tightly entwined work with UPS and FedEx or push their own truck delivery system which is probably why they're putting effort into drone delivery. There's technology out there for "smart shipping" which will pack stuff into a delivery vehicle which can cause the vehicle to be packed based on the route that it will take. This being optimal as you can reduce the volume required since you no longer need to have space for the driver to get into the storage portion of the vehicle and search through the packages for the one to deliver. This means either smaller delivery vehicles or more packages in the delivery vehicle. It also optimizes the delivery process a bit more for automated warehouse to door shipping.
To do this you need to be able to stock your vehicle to a reasonable fullness which means packages need to sit in a staging area for the loading dock until some combination of time and volume quantity has been achieved.
Amazon is probably using the drone delivery right now because automated vehicles are still a restricted technology or they may be assuming that UPS and FedEx and the like are also at least doing some element of research into automated delivery systems in order to not be obsoleted by it.
My feeling is that picking and shipping, while done by humans, are going to be solved by the same technology. The big issue with picking is making sure the goods aren't damaged by the robot hand when selecting various goods from within a container. Shipping should be a quick to solve problem once the picker problem is solved as the part about picking items out of the container to pack into a box is solved and at that point its a computation problem of the dimensions of the products and optimizing the order to package them as well as any packaging.