Youtube has the right to stream those videos, because they pay the RIAA for a streaming license, which is a thing that music licensing agencies have provided for many years now. In many cases, the videos were uploaded by the recording studio itself, and in those cases presumably Youtube isn't paying a licensing fee because the license is implicit in the upload (but note that many videos uploaded by the artists themselves are not technically theirs to upload because the actual rights were held by the recording company, hence the reason Youtube needs the agreement with the RIAA).

The DMCA take-down exemption you are thinking of is a separate protection, that applies to Youtube taking down videos that aren't licensed by them containing music that isn't licensed by the creator of the music.

The statutory damages would apply in the former case, such as if an artist uploaded their own music video and Youtube didn't have the broad RIAA license. This happens a lot more frequently than you would think.