They don't have an exclusive right to scan in books. First, because such a right could only be granted by either copyright holders, as an exclusive license (which would also necessarily mean that scanning the licensed book was not infringing) which wouldn't matter to rivals because the rivals can ignore such a license and rely on fair use per the lower court's decision. Second, because the only other way to have an exclusive would be if there were literally only one copy in existence and the library that owned the copy refused to grant access to anyone else, and that is frankly, pretty unlikely.
The reason that they may have an effective exclusive is simply that it's an expensive pain in the ass to scan all of this stuff in, and there's little money in it, so who else would want to bother. But the disinclination of third parties to compete with Google because it's hard, likely minimally profitable work, is hardly Google's fault.
The point is, they took on a project that violated copyright on a massive scale. They want to claim that it's fair use
No they didn't. Fair use is by definition not a violation of copyright. And so far this has been determined to be fair use, and with the Supreme Court refusing to take up the case, there should be nothing else to say about it.
I think it's only fair the LoC get a full copy of their book index.
That would be nice, but they're under no obligation to give a copy to the Library of Congress if they don't want to.
Frankly, I think its good that such a thing exist in society. But it's not good that a private for-profit company can take it on themselves to do that for their own gain.
Why not? Certainly the government should be doing this sort of thing; as an attorney it always bothers me that there's no public alternative to Lexis and Westlaw. But that the government could do it and should do it doesn't preclude private entities from doing it too, as a general rule.
For instance, the government builds most roads. But nothing stops a private company from building a private, for-profit highway, so long as they can get the land without government assistance and afford to build a safe roadway on it which complies with various regulations. It's just such a hassle that it's rare.
Honestly, it's strange that laws suddenly stopped applying because it was on a computer.
Copyright law applies. Google simply hasn't acted illegally is all. Google books is no different than if someone made analog xeroxes of lots of books, manually compiled a master index of everything in them, and took requests by phone to tell people what books matched various search terms (possibly with a specific sentence or passage read aloud over the phone to lend context to the result).
Computers make this practical, not legal.