They provide the most service to those unable to afford help. Also, they mostly provide the intangible service of raising peoples' education. That's why they don't make money. People only go to college because they think they're screwed if they don't, but almost nothing else education related has that amount of leverage in the peoples' decision-making.
Fundamentally they don't make money because they don't charge, and also don't sit around trying to think of ways to make money (while still remaining true to their core values and purpose). I don't think we should assume it's an insoluble problem. Once one has an audience and does something useful, monetizing can often be tricky, but is often possible.
Just as one example, what if libraries had some ads somewhere? Then they could be provided for free to poor people but also gain some revenue. (And you offer premium membership with no adds for a yearly fee.)
If that would work (I don't know), then the question would become: should we charge people taxes so that poor people don't have to deal with ads at libraries? And I think the answer to that would be "no" -- if you want to help poor people see fewer ads you could pay money for that yourself without the tax system (e.g. the libraries could directly accept money and the more they get paid the fewer ads they show according to a formula).
> Just as one example, what if libraries had some ads somewhere? Then they could be provided for free to poor people but also gain some revenue.
Perhaps when people borrow a book they are only given the first four chapters; they have to promote the library to get access to the next three chapters; and they have to pay to get access to the last four chapters?
I doubt freemium models will translate well to libraries.