Not really. They have to do something, or they become liable. If youtube decides that something is fair use, and a court disagrees, then they are on the hook for damages. They'd have to pay a lot of money to copyright lawyers, only for the chance of having to pay damages.
And, you know.., The same libertarians, who are now attacking youtube for not going full feudal, would be absolutely outraged if they did fight for fair use. It's stealing property, as far as they are concerned.