that's the desired end target, yes, but we're realists. we view our role in the world to pull the levers of power we have access to to get to that end target. libraries are repositories of free knowledge, tools, and entertainment. you'll encounter a lot of anarchists arguing that we must immediately disband the police, but almost none arguing we must immediately disband the library.
so in our current place of existence public libraries are proof positive that the collective sharing things is a good and natural positive and that charging money to just exist isn't a positive. librarians are also as a whole very dedicated to helping people develop knowledge and skills that help them navigate the hellscape we're all forced to exist in.
you say no true anarchist would support a public library, i say no effective anarchist is that dedicated to purity.