I doubt they can. Once they learn about something infringing, they have to remove the link “expeditiously”. A proper DMCA notification is not even strictly necessary. Even so, the DMCA specifies what information a proper notification should contain. An email address is sufficient ID. ->DMCA
The point would be to ask that data before you could make a DMCA notification, but I agree that you could not fix this way a bad written law.