Running then with the library example, who then would be inclined to fund and run libraries in local communities across the nation (where companies may not be heavily invested or even able to afford such efforts)? In the libertarian stance, what would the alternative here be?

Don't get me wrong - I generally find the libertarian mindset quite agreeable but there are certain examples where I'm unable to reach the conclusions truer libertarians do, and this would be one of them.