I was never a fan of my public library. When I was young, I found the books in my own home to be all I usually needed. Of course not everyone has a home chock full of books and reference materials, so I was very lucky to have that.
When I was older, I found that the library still did not meet my needs. First, the technical books I wanted were seldom available. I might be able to request them from a larger library, but that took time. I also had to return the books and with most of these, I wanted them available longer. So my books came from bookstores, physical at first but later from the Internet. Again, of course I had the money to do that and not everyone does.
Because of that "not everyone does", I absolutely support my tax dollars supporting libraries. However..
Most library activity is entertainment, not research, not knowledge. It's still difficult, even with SAILS, to find good technical books. Romance novels, detective stories, sure, hundreds of those. But the tech side is weak at best.
So my question is this: would the general public support libraries if all that entertainment went away? I don't think they would, because most people don't really care about anything else. They'll SAY that they think their library is a valuable resource, but I think that's mostly lip service.