Erm, publicly funded libraries do pay: they buy the books. With pbooks and ebooks they pay for the number of copies that are in borrower's hands. Even libraries that accept donated books (and it seems as though rather few will do so for lending) have indirectly paid for those books via the donor.

I also don't understand how governments figure into the picture. What is the difference between a public library (government funded), a university library (partially government funded), and a private library (no government funding)? For that matter, what is the difference between a public library and a used book store? Sure, money changes hands in the latter case. Yet none of that money goes to the author or publisher.

As for competition in lending, it already exists. Universities will gladly offer you a library card (for a fee). Many organizations offer libraries for their members. Everything from community centers to grocery stores will have a free book exchanges (which is really just lending with informal bookkeeping). The main difference is that public libraries are open to everyone while those other libraries are exclusive. And that openness to all avenues of society is what makes public libraries a good thing.