Take down => Come up somewhere else.
Forget Youtube for a moment. Sure it's the dominant player, sure it's the 800lb gorilla, but think more generally for a second. Lets suppose we succeed in forcing Google to take things down, and make sure that they never come back up again on its service. Then what?
Google loses traffic, advertisers go elsewhere and someone else starts a video sharing service hosting all that tasty content. The only reason they wouldn't is if they were subject to the same laws, but then all we have is a case where we can legislate all the TakeDown=>StayDown we like and it just means TakeDown=>ComeBackUpInRussia*
As long as there's somewhere that's prepared to host this stuff, then it comes back up, end of. That's how the internet works. It doesn't care about political or geographical boundaries. Anyone can attach a machine to it and put what the hell they like on there.
If we really don't like that, then we have to start talking about "breaking the internet" again, and deliberately making things that are technically working as designed unroutable - which is rather a slippery slope to go standing on.
So, really we're asking:
"Is it worth breaking rights-holders business model, potentially to the point of driving them out of business entirely, in order to preserve the operation of the internet as designed?"
For what it's worth I think that's a good trade, but as a raging anti-censorship all-speech-should-be-free-speech and damn the consequences type I imagine there's lots of perfectly sensible people who won't agree with me. That said, I don't think it's terribly controversial to suggest that when there's really no legal recourse (because there's no possible way to enforce it internationally) then we're back looking at technical recourse such as blacklisting - and those have a bit of a habit of being abused, so this needs to be taken both very seriously and very carefully.
* Or elsewhere, for that matter.