I would much rather billionaire types take on large issues that governments don't want to touch, like space flight or specific diseases in foreign countries, than something already in service like libraries. Maybe I just have a bias due to having a library just up the street from where I live currently, but I have never not had access to an adequate library in my life time, and only see room for marginal diminishing…
Visit a library in a low-income area (where people need them the most) and see the kinds of computers people use to hunt for jobs, do their homework, etc. I've been to a library that was running windows 2000 on TWO ancient Dells with with CRT monitors in 2015. And the two computers they had were being used by adults and students. There is so much to be gained from improving libraries.
Libraries provide so much public good: giving the public internet access, allowing people to read classics and better themselves for free, and giving some quiet space for those wanting to study (and not have to pay for a $5 coffee).