[sakoht]

Setting aside the evils of the DMCA itself:

Seems clearly that if youtube authorizes downloads, then people shouldn't put things on youtube they don't want downloaded?

The takedown notices should be going to youtube. But of course they want to use youtube to promote themselves.

[pteraspidomorph]

If the video is viewed directly on youtube, the majority of viewers will also be served ads which generate revenue that is shared with the copyright holder (note how youtube identifies the music in the video and who the copyright holder is below the description of the video). So it's not about not wanting it downloaded, it's about not wanting the ads circumvented.