Videos on Drive
Some users report Videos uploaded from Drive and embedded in their Site do not play, particularly if not signed in to a Google Account. This seems to be a common issue for Google Drive users. It does not seem to be a Google Sites embedding specific issue but a more general Google Drive issue.
A quick search yields a lot of reports and potential solutions.
Until Google Drive fix this issue, I think Sites users are better advised to upload their videos to Youtube or Vimeo and embed from there.