This is not about false claims, it is about site owners learning their rights. There is an ocean of stolen content out there. Site owers simply have zero way of controlling all of it, but DMCA actions are one way (in the states atleast) of addressing it. More and more content owners are getting clued in on how that process works.
However, there is still no direct method for addressing the billions of stolen pages setting on foreign servers. For that - we needed the protections that SOPA would have provided, but it was muddled with the "censorship" word and everyone remained ignorant and sided with Sergey Brin on the issue.