You had that right, up until the point you agreed youtube's TOS/EULA. And you are still free to bring a suit under the DMCA ... except you also agreed yourself out of that too.
Well, the DMCA has something to do with in, in that the DMCA approach of adopting a safe harbor process rather than a mandatory process is exactly what encourages businesses to make other arrangements that differ from the DMCA process if by doing so they can reduce their costs and satisfy the key players from whom they would be concerned about litigation in the absence of the safe harbor, while neglecting the interests of parties -- e.g., most people that aren't big media companies -- to whom they would have no or insignificant liability without the safe harbor.