From everything I've seen the related media type for video is only going to pull in for Sitefinitys video player. Although I know the video widget can handle YouTube videos you can't get one into the library (at least not without customization).
You could handle this pretty easy though by either:
1) Create a new short text field in your module where someone can just paste the entire embed/iframe code and then pull that into the widget template.
2) Create your own dynamic module for YouTube videos which would be set up the same way as above just in a separate module, then pull that in via related data.
You could also have the iframe code set up in your widget, so the only thing a content editor would need to add in is the video code itself.
src="<%# Eval('YourShortTextField') %>" frameborder="0" allowfullscreen></
Hope that helps!