And therein lies the rub: yes, Google/YouTube might remove copyrighted material whenever requested, but by that time, the cat's already out of the bag. With tools for downloading video from YouTube, by the time it's removed thousands or even millions of copies might already have been made.
These services must devise methods to prohibit the infringement in the first place. How they do so is up to them, but until they do, they're providing a platform for committing crimes. At the very least, such civil actions as that discussed in the linked CNET story should succeed.