Given that the private sector does seem fully capable of providing libraries to those who want them, why do we subsidize them?
The question is rarely as simple as "could the private sector do it?"
Healthcare provision is rivalrous and excludible, ask any poor American, doesn't mean it shouldn't be universally provided by the government.
Education is rivalrous and excludible, look at private schools - doesn't mean it's wrong for there to be state schools and laws making children attend them.
The only thing that matters is whether the public benefit outweighs the burden on taxpayers.