We could start by not making censorship the default: If a notice is received, notify the poster of the video that it will be removed in e.g. 24 hours if they don't submit a counter notice. Then it comes down after the grace period if they don't identify themselves and file a counter notice consenting to the jurisdiction of at least one U.S. federal district court, but if they do then the content stays up until a court says otherwise. And given that filing a counter notice is a de facto invitation to be sued, infringers who submit them improperly will have to answer for it in court, which should reduce the number of people willing to do so. On the other hand, in cases like the one in question here, the facts will have to go through the copyright holder's actual lawyers and a federal judge before any censorship occurs, which should do a good job stamping out fraudulent take downs as well.

In reality this is how the legal system works for almost everything else: First you ask nicely, and if the parties can't agree then you go to court. But in most cases the party in the wrong (at least in theory) knows that they'll lose in court and so concedes immediately and everything works as it ought to without taxing the court system.