I'm not sure why people seem to forget the other parts of the DMCA, one of which is the safe harbors that protect sites from being liable for content by users that infringes on someone else's copyright as long as they take it down. Without it, sites would just be sued directly for the copyright infringement and they'd go away or stop allowing people to post anything. Giving a clear process for takedowns (including counter-notifications) is better than the alternative, even if there is some abuse.
In order to deter the false ones, they just need to actually go after those that perjure themselves claiming they're the owners (or their representatives) of something they don't actually own instead of leaving it to someone like Google to enforce in civil courts.