Often, when they do, it is either:
(1) Because of state funding specifically for rural counties, which amounts to a revenue transfer from urban counties, or
(2) Because the rural town is in a county with, and deriving revenue from, a city, funding county libraries in the rural town as well as the city.
Even more often, tiny rural towns don't have public libraries, and you have to go to a neighboring town or the nearest city.
I think your bias is showing.
I have lived in rural, suburban, and dense urban areas in my life (currently high density urban), and there is a bias on HN toward the idea that rural communities are unsustainable on their own and mere leeches on the city dwellers. That's likely true in some places, but not universally. Sustainable rural communities have exports of resources, housing, and/or tourism, that outweigh their cost.
As the parent comment describes, one model for remote communities is volunteers who work at the library because they care. The cost of everything is lower -- labor, food, matetials -- so what looks too expensive for a town might actually be fully sustainable.
The sustainable rural community is exceedingly rare - most are propped up by ag subsidies and state aid. For much of the central part of America, there's no housing benefit or tourism to speak of, just resources (corn/soy in general, maybe livestock).
You then counter by saying that's wrong and actually small towns have a small library that has a crappy selection but they can get you any book you want if you can wait.
Pop quiz hotshot: Where do you think there is enough built up infrastructure, wealth, and surplus labor capacity to have a large centralized store of books and to organize and distribute said books?
Probably just another small town, am I right?