I think that remixing should be allowed. The music usually sounds quite different from the original. For example, in the video a clip of Girl Talk making a remix on his laptop shows him just using beats from a song which doesn’t have any resemblance to the original at all. Artists should feel proud that other people like their music so much that they want to use it to make their own songs.
Remixing can be related to the field of architecture. Architects sometimes draw inspiration from other architects to create their designs. This could be looked at as a mash-up of various buildings. I know that I sometimes bring in parts of other buildings for inspiration in my designs.
If I were to use parts of the RiP: A Remix Manifesto video for my remix, I would possibly use the parts that the narrator is talking about what remix is and what the controversy of remixing is.