I suspect that it could have but it wanted to take full control of its digital assets within the confine of their own website. You see, allowing other websites to embed your content inevitably means you have to pay them a fee for being an affiliate. Why not keep everything in-house to maximize profits?
I think we also went through a brief period a couple of years ago where companies were worried about their videos being embedded everywhere. Even if they were embedded on random sites with the ability to accrue ad revenue, I just don’t think MTV wanted to deal with it. Could be wrong.