Think about it in purely economic terms; the way the average book buyer will look at it. I spend more than $80 a year on books. The obvious question is, "Will that $80 a year for Amazon Prime give me access to enough content to be a better financial choice?" For me personally, as an academic, the answer is no, but I am the exact opposite of the average reader who will do that math and come up with a "Yes." It's like a movie fan deciding on Netflix. The cost is so affordable compared to what they usually pay it becomes a financial no brainer. This is especially true in light of the commodification effect of ebooks: most readers only want to read the content and dont care where it comes from.
Libraries can hope that Amazon discovers in the long run that this model is not affordable for them, or that authors and publishers will do what Hollywood did a few years back with Netflix- ask for even more money and limit the number of titles available. Netflix has seen there catalog shrink even as licensing goes up, because Hollywood wants to cut them out of the picture. Hollywood wants to cut libraries out of the picture too. I think publishers want to do the same as well. I would not be surprised to see a Big 6 lending library emerge in the next few years. It makes sense for them to either go it alone or team up. They could undercut Amazon Prime by a few dollars, and offer "patrons" exclusive access for a simple yearly or monthly fee. That way they can increase their profits, control their content, and lock their consumers in. Guess who doesn't fit in this scenario? Libraries.
There is no way of knowing if this is a viable business model, but it will challenge libraries relevance for the next decade or so. And unlike Amazon, we just don't have large amounts of money to spend experimenting with potential models.