I suspect the reason why they don't is because of legal concerns. It seems like the vast majority of the videos uploaded to youtube are videos that people recorded themselves, whereas the vast majority of mp3s that people share in almost any context are pirated music. There are probably certain small communities where there are plenty of user created mp3s shared, but it doesn't seem to be the case for a sharing platform of any significant size.

Grooveshark manages to get licenses for all these "gray" mp3s after the fact, but I doubt youtube would be able to do so as easily considering their larger market position would be more threatening to the labels.