Out of curiosity, has there ever been a case where a government has demanded that Youtube remove a video and it has not done so? I may be wrong, but I actually believe that they
are fully responsible for the content that they are carrying. They
do actually go to considerable lengths to remove content that they think is in violation of their terms of service as promptly as they can. I don't believe they are required by any law to
review every submission for possible problems; only to remove content that they are aware is not compliant. I believe they do that. If they don't, I'm sure some enterprising lawyer could fashion a lucrative class action suit. It's not like they aren't sitting on a mountain of cash.
Just to be clear, they are not a common carrier and I have never seen any evidence that they claim to be one. I don't think they have to both ways. I understand (and have sympathy with the idea) that it would be nice if they were, but I really don't think they are.