My issue is definitely not with the copyright of the material, it is to do with the "perceived" right of the network or the recording company to veto the posting of videos on YouTube.
If a network or recording company has copyright in a given song, or a given recorded performance of a song, it's not "perceived" right--it's a legal right.
Once again, why would it be more beneficial to the record company or the artiste to "pull" the video rather than let it remain and gain exposure?
That's within the judgment of the copyright owner--the copyright owner of the song, or of the particular arrangement of the song, or of the recorded video. Each of those (potential) owners makes his/her/its own judgment of the relative benefits, based on his/her/its own advantage, which may or may not coincide with the perceived advantage of the other owner(s) of copyright in the particular video. The performer (who may not own the copyright in the video) might want the video there for publicity or ego reasons; the owner of copyright in the video may or may not agree; the owner of copyright in the underlying song may or may not agree.
I am not a copyright attorney, but as I (mis?)understand copyright law, any one of those owners could rightfully demand withdrawal of the "offending" item, even if the other owner(s) wanted it there. Always assuming that appropriate fees had not been paid.