I don't think anyone is arguing libraries aren't worth having. The argument is how to spend private money.
If libraries are underfunded, that is a problem you need to push through public channels. In the same vein as why a private company doing infrastructure makes no sense, if we can demonstrably show the benefits of having libraries (and we can) that breaks down the most fundamental barrier to public funding of anything - justifying the expense with demonstrable results.
The government has a hard time investing in space flight because without a demonstrable major goal like beating the Commies it looks to the uninformed like money down a black hole. It is really hard to justify, no matter the degree of information availability, bleeding edge research on the public purse because you have to explicitly detail it isn't a guarantee on results for any given years investment. In politics where the mood changes based on yesterdays news, few can weather a bad year if they are putting large amounts of money into important research.
Libraries, though? We got statistics for that. We are well versed in building and staffing libraries. They are functionally infrastructure - and that is supposed to be what government is good at. If you live in a country where it is not good at infrastructure, you got bigger problems to deal with then.