It's past time for state and local government to require large online retailers such as Amazon to provide a means of collecting shipping materials for recycling and reuse... This segment of our economy badly needs some gov regulation... For the sake of the planet
There is a way, that's the point of the entire article. They put it in the recycling.
If you are suggesting shipping the stuff back to Amazon, you should think about that a bit more - the amount of energy you'll use to ship it is an order of magnitude more than the amount you'll save.
You would be harming the planet in order to make it seem like you did a good thing.
For at least some types of retail products, it seems like a system of standardized, reusable containers could be workable - the same Amazon, UPS, USPS delivery vans that bring a constant stream of packages to my apartment building could collect and return them without much additional energy expenditure.
Possibly using something comparable to the German Flaschenpfand[1] system? I'm unsure of how to develop a suitable packing system that would be machine-friendly, possibly wooden crates packed with shredded cardboard packing. The lid could be press-in cardboard, something like that used in tinfoil food storage boxes, but reliant on the cardboard bending to get past the lip and into the region where it is secured. Alternately, a suitable slide-in wooden lid with a stopping peg inserted and sealed with a wax-type substance could be used (as per [2], with the addition of a sealing peg setup). Sealing wax would probably be used in the cardboard variant also.
Amazon handling recycling is not going to have any fewer problems than local governments doing it. The problem, ultimately, is that there's no way to responsibly extract value from recycling this sort of stuff. The choice is burying it or selling it to someone who will irresponsibly extract the value.