Treating books published within the past year as "premium" content and making the rest available for a flat rate could work. Cable TV works more or less like that in the US. You could have a higher fixed rate for particular areas of content (e.g. I'll pay a bit more to have access to new SF, but I won't pay a dime for romance if I can help it).
I really don't see the US government being willing to get involved in this at all, though. We can barely get PBS funded, and that's TV, which Americans like. Municipal libraries have enough local presence to manage to scrape up some funding each year, but it's a tough fight every time, as local librarians can tell you, and that's an institution with a long existing tradition.
As for the censorship issues Steve Jordan raised, I don't think the US government is stopping anyone currently from buying "undesirable" books in bookstores, or even borrowing them from a library. They like to track what people buy and borrow, though, and I could easily see that being a factor in an online library. (Some public libraries refuse to keep records of what their patrons borrow, for exactly that reason, but I think an online library, whether for fee or publically funded, would be pressured into keeping such records and would be forced to divulge them on demand.)