Restricting and Filtering YouTube Videos
Web Safety has the ability to enforce Strict and Moderate YouTube filtering as described in this Google Support article.
In order to force different level of YouTube filtering navigate to Admin UI / Policies / Desired Policy / Filtering Rules / YouTube / General and select what level of YouTube restriction is required - None, Moderate or Strict as indicated on the following screenshot.
To test the actual enforcing of the restrictions, you can visit the Content Restriction check page on YouTube. The response page should indicate the type of restrictions enforced.