> They don't have to put the video back up or do anything when someone sends a counter notification, but that means they have full copyright liability for that video. They can now be sued directly for violating copyright for each copy they made when someone viewed the video prior to when it was taken down.
Anybody claiming to own the copyright. It doesn't matter if the claim is legitimate; "safe harbor" means the time and effort wasted on copyright problems is limited to hiding/restoring videos and forwarding a few legal notices between the claimant and alleged copyright infringe channel. The expensive parts like defending against potentially frivolous claims happens directly between the parties involved.
Sued by whom?