Note also that YouTube has specific requirements for use of their embedded viewer and API. You can't just use it any way you want. In particular, if you play the audio of a YouTube video through their API but hide the video, you'd probably violate at least one of
8. separate, isolate, or modify the audio or video components of any YouTube audiovisual content made available through the YouTube API;
9. promote separately the audio or video components of any YouTube audiovisual content made available through the YouTube API;
14. use a video player smaller than the minimum video player size set forth in the YouTube API documentation and specifications.
If you're really just embedding the video a good-faith effort to show the video in a new context, you're probably okay (but check the terms first, just to be sure).
In practical terms, a copyright holder might agree to this because YouTube includes advertisements with popular videos, so allowing a video to be embedded might mean additional ad revenue for the copyright holder. Alternatively, if a video isn't popular enough for the creator to get ad revenue share from YouTube, getting more views moves it closer to that threshold.
The restaurant case is a bit trickier. In general, when a copy of a copyrighted work is transferred to you, the right to public display of that work is by default reserved by the copyright holder. (In the U.S., this is covered by 17 USC §106(5), "to display the copyrighted work publicly.") The right to view a public video on YouTube is offered to each user of the service. In general, I would not assume that each person in a crowd of people looking at a YouTube video on a public screen are users of the service, but I may be wrong. If such a crowd are not all "users of the service" and in fact you, the restaurant proprietor, are the only true user, as the interactive operator of the computer, then you would be performing public display for the benefit of your patrons, which is a right under copyright that has not been licensed to you.