10 seconds is the marginal cost /if you're going to the store/. It's entirely possible to make zero regular trips to the store under normal conditions. I'd use the heck out of a "cause more cat litter to arrive in a day or so" button, in fact I have a bookmark on my laptop for just this use. Mobile apps are (imo) universally horrible for this kind of directed action.
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.
Personally I think smartphones will never be good at known-goal interaction. They're general-purpose devices with a hilariously low-throughput interface that need to accommodate low-tech users. You're always going to have to navigate a bunch of screens to get to any given function because the real estate is so small.
Could apps be better? Sure. Can any app get as good as a button that's already right there when you realize you need a replacement? I don't see how.
How do you get your food and other things? I'm genuinely interested. I've lived for periods where I ate out every meal but I still ended up in a store now and then.
Safeway or another chain probably delivers if you're in a major metro. Some smaller cities have local stores with online ordering, too. And, of course, with some more trouble, you can get groceries delivered in just about any populated area if you have access to a phone.