Public libraries are really a thing of a past generation: they never really have the books you want to begin with, their search engine's awful, it takes forever to find and it's often still checked out. When you factor in opportunity costs of getting to and from the library and the late fees you'll probably accrue, a kindle's probably going to be worth it if you're a heavy reader.
Like another poster here, I live in a highly educated area. We all use the library, and we all (my neighbors and us) take our children to the library. =-)
It's true that our library system shares some books, CDs, and movies across the library system (usually for less popular titles). It's no biggie because I log into the system from home, put the item I want on reserve and then--after I get the email announcing the item's arrival--waltz in to pick up the book/movie I requested from the self-serve shelf.
It did not take long to get The Goldfinch--a highly popular book now--after I requested it.
Recently, I read that a good number of people now read both paper and digital books. I fall into that category, though I read a bit more on paper than I do on digital.
The cost to get to the library is zero dollars because I can walk or bike. Sometimes, I do drive a car if I am getting a lot of books. We're allowed to check out more than 60 items at a time.
Another lovely advantage to the library is the accidental discovery of a title, essay, topic that you may not have stumbled across in a digital format. I checked out two books of essays the other day and found several essays on topics I'd been studying. This was luck and a lovely coincidence that I rarely experience when I download books...because when I download a book, that is all I get. There are no other books on the digital "shelf" unless I already know to put them there.