Sure, but in a community where there wasn't an actual public library, I do believe one would inevitably come into existence as the effort of the community, in the same way those tiny libraries do. "People have spare books; community centres exist; so why not put the spare books in the community centre?"

In the alternate reality where public libraries didn't already exist, I suspect the law would make it illegal to loan books. It would be a direct response to these small informal libraries trying to share books and "violate the license."

But you'll be constrained to the old spare books of that particular community. Public libraries have a much broader selection, and they also buy new books.

Also I'm not sure they'd actually come into existence everywhere, even if the internet is a poor substitute for a library it is cheaper and may be enough for many.

But only a certain kind of book; government approved books.

Definitely nothing interesting.

Very possibly, yes. The original modern libraries in the west were effectively subscription libraries, as in, they weren’t free to use, you had to pay to become a member. It was only later that they were run by the government. That said, there is a vast gulf between them and a little box people put outside their homes with some books inside.

Further reading: The Library Book by Susan Orlean

Except communities that under-privileged will be unlikely to have the bandwidth or know-how for activism at that scale.

Unfortunately reality does not "inevitably" progress towards utopia, any more than the life of any individual does.