> Probably most of the relevant functions of a library can be accomplished by other more efficient means.

Two points:

Many of those relevant functions were picked up by libraries over the years due to the lack of, or cutbacks to, other government services.

Some people prefer, or can only use, services offered through libraries because they offer them with few, or no, questions asked. That is rarely the case with dedicated services, where funding often has more conditions tied to it.