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.
I was having this problem myself using Chrome and I discovered that by going to chrome://settings/content/cookies and click the 'Add' button next to 'Allow'adding [*.]google.com to 'Allow' fixed the problem for me.
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.
**** UPDATE Oct 2024 ****
Hopefully the new Google Drive video player described here https://workspaceupdates.googleblog.com/2024/10/modern-video-player-in-google-drive.html will resolve these issues. See also https://workspaceupdates.googleblog.com/2024/10/modern-video-player-in-google-drive.html especially the part:
"Tip: To play videos in Drive, you need to turn on third-party cookies. Learn how to turn cookies on and off. "
An example of a video on Drive embedded using the Embed by URL tool
https://drive.google.com/file/d/0B9K6ilf0g9SSMzQ4ODNjYzctYjczOS00YzNlLWE3YTEtYjViNTZjMmM2Njg2/view?usp=drive_link&resourcekey=0-9AJqrJT7SMaUdgTQg5WJMw