Which is exactly what they are doing at the moment. When notified of an (for the sake of brevity) infringing site, they de-list it.
That's not whats being asked of them here though;
There's a certain video of Max published on the net.
At the moment, if it appears on ABC.com/Max he needs to tell them about it, which is a lot of effort because it may well be published on XYZ.net (or even ABC.com/Max2)
What he's asking is that rather than having to notify them of every page (which is an insurmountable task really) that Google detect the video in software and auto-delist pages. That's a very, very different proposition and is never going to be 100% accurate (point of fact, do you err on the side of caution and risk false positives, or risk false negatives and being sued?).
There's a big difference between monitoring YouTube and monitoring everything else. Even if you were to use video fingerprinting that won't be 100% effective (I could make dramatic changes to the video whilst still ensuring the original was the primary focus).
It's not impossible but it's not a good idea, and is something that will get misused should it be implemented.
The idea that people can only find things through Google also ignores the wider picture. It may not be in Google, but if I link to it on Facebook, everyone on my page will know the URL. That's part of how videos go viral in the first place!