They could probably mitigate the work involved in reviewing videos by using a trust rating system combined with semi-random manual reviews.
In this system, people who have uploaded a relatively higher number of videos with no copyright violation issues, would have a relatively higher trust rating. Their videos would have less chance of being flagged to be manually reviewed. As soon as any copyright violations were detected, their trust rating would go back to zero.
Videos which were flagged to be manually reviewed would go in a review que and would take significantly longer before they were released for public viewing as a result. This would give people a reason to aspire to a higher trust rating.
Basically, a sandboxing system similar to the one that Google are reputed to use with their search engine.
In this system, people who have uploaded a relatively higher number of videos with no copyright violation issues, would have a relatively higher trust rating. Their videos would have less chance of being flagged to be manually reviewed. As soon as any copyright violations were detected, their trust rating would go back to zero.
Videos which were flagged to be manually reviewed would go in a review que and would take significantly longer before they were released for public viewing as a result. This would give people a reason to aspire to a higher trust rating.
Basically, a sandboxing system similar to the one that Google are reputed to use with their search engine.