With vinyl records, and cassettes, and compact discs, we listened to music on speaker systems and on headphones. Take away the physical media and replace it with files on a computer, and we're still listening to music on speaker systems and on headphones.
But books? We never consumed books through something analogous to television sets or speakers. We held the actual book in our hands, and read the printed pages with our eyes. Replacing physical books with digital books is, in my opinion, a significantly bigger switch than moving from compact discs to MP3s or from DVDs to MP4s. The entire experience has changed, not just how the content is stored.
My mum is a member of a library, and they have ebook which she can download and read on her laptop. However, she can't read it on the Kindle, as they are distributed as DRM protected ePub's.
To put it on her Kindle she would have to remove the DRM and convert to mobi... But she doesn't need, or even want, the DRM removed.
Also, second-hand books. I could very easily get hold of a Dan Drown's The Lost Symbol for £0.50 in a charity shop/second hand book store. But it costs £4.96 to get the Kindle edition. There's also no way to trade/sell/give Kindle books to other people.
Removing DRM but then deleting within loan period- or pirating and donating £0.50 to charity - for ebooks seems morally equivalent to me. But it's illegal/break of copyright.
Personally, I think it's not really the demise of the Public Library but more the demise of our society due to the lack of proper priorities.