My question/complaint is basically: rather than put all this effort behind a single-purpose button, why not work on reducing the friction for using a mobile app for the same purpose? There's no reason you couldn't have, say, a button on your phone's home screen where pushing it causes cat litter to arrive. This would be about as convenient, and way more flexible.
Edit: and the answer is pretty clear, I think. If people actually go for this, it'll be a massive advantage for Amazon. It essentially banishes competition permanently for people who use the button regularly. It'll also make people much less price sensitive. So I completely understand why Amazon would try to push this. I just don't understand why anyone else finds it interesting besides that.