You're right that the domain registrar does not own the server but they do however own the domain which you point to your server. The domain registrar reserves the right to retract their domain if its misused such as pointing to illegal content. What counts as illegal content is sometimes vague but thats another issue in itself.
I do agree with your sentiment that the DMCA notice should be sent to the site first but in the end, hosting providers and domain registrars still have some rights and policies which you have to comply with.
