The usual strategy is to create an account called Anonymous Gift Vidder or whatever on YouTube or Vimeo, then choose the option that keeps the uploaded vid out of the search results (for YT) or password protects it, while putting the password in the body of the AO3 post (for Vimeo). There's a full explanation here.

Anyway sometimes no one ends up creating for this media option, so it's not a huge deal either way, but I might if it was a thing and I always like to see more things included.