1) Eliminate legal liability and risk of lawsuits. Technically all they need to do is comply with DMCA requests, but YouTube would rather not be sued by a major record company or film studio, even if they could win the case.
2) Become friendly with major content creators (record and film labels, television studios). YouTube wants a good working relationship with these companies, to ensure that these companies post their content on YouTube. YouTube really wants the views they get by hosting SNL clips and Beyonce music videos.
3) Actually help the independent content creators deal with copyright claims. This is a very, very distant third.
Because of these priorities, YouTube has decided their best course of action is to offload all of the work on copyright claims onto the independent content creators, in order to keep the big media giants happy. After all, NBC doesn't need YouTube, but a solo channel with 50k subscribers absolutely does.