First, I would comment on the question and notify the OP that the embedded video is no longer available. Maybe they know of an alternate media, or article that can replace it.
If OP did not respond, I would try to find an alternate media source — another video, image, or article that illustrates their point — and edit that into the question.
If I could not find a suitable replacement, I would check for an archived version of the video at the Internet Archive Wayback Machine. (The Internet Archive Wayback Machine can make backups of YouTube videos.)
If there's no replacement or archive copy, and the video clip was not essential to answering the question, I would edit the question to indicate that the embedded video is no longer available, and leave it at that.
I am positive there have been other questions with their clips struck down, where the missing video makes understanding the question itself problematic.
If the video clip was essential to answering the question, in that without it, the question lacks the required context or details for it to be answerable, I would vote to close the question as “needs details or clarity”.
For example, this question: Are Bollywood actors actually filming 30 to 40 films at any one time? — OP cited a claim (made in the embedded video) that was essential to answering the question. The YouTube video is no longer available. I notified OP that the embedded video is no longer available and asked them to add an alternate video, but they did not respond.
I tried finding alternative sources for the claim, but I could not find any. I tried googling it and was led back to the question. (There's even a comment on the question doubting the OP's claim.) Since the question now lacked context or details for it to be answerable, I voted to close the question as “needs details or clarity”.
Arqade has a similar policy where if the media is no longer available and there are no alternatives, the question is closed as “needs details or clarity: How should a Game Identification question that no longer has a valid artifact be handled?
When the linked resource is no longer available, and we cannot easily
locate an archive, mirror, or reupload, closing as 'Needs Details or
Clarity' makes the most sense.
[…]
There's also a question of usefulness for broken-link [questions] - if
the source has been taken down entirely and no such backup exists -
it's not really worth keeping on the site - it won't be useful to
future readers. […]
So yes, 'Needs Details or Clarity' is the correct close reason in this
case.
Addendum to address comments:
I would agree with most of your answer. However, I don't think it's correct, either legally, or morally (ethically?) to re-post or re-embed the video clip from the wayback machine. The clip was struck down as a copyright violation, presumably by the copyright holder. What we, individually, or as a group, think about this act (even if we are sure it should be allowed (in the US anyway) due to 'fair-use' doctrine, the copyright holder struck it, and if we replace it, then do we not have a potential liability?
We, as regular users, are not responsible for enforcing the removal of any embedded videos that may violate the legal rights of the copyright owners.
YouTube, Internet Archive, and SE have their own Terms of Service that specify how to report any content on their sites that infringes any laws. They are the ones who have to deal with it, not us, the users.
If the copyright owner also wants to take down the alternative YouTube video or Internet Archive mirror, then that is their prerogative. We should not worry about any legal issues that we are not involved in.
