In all cases the HTML looked something like this:
<iframe style="max-width: 100%" width="560" height="315"
src="https://www.youtube.com/embed/FgxwCaL6UTA"
frameborder="0" allow="autoplay; encrypted-media" allowfullscreen="allowfullscreen"> </iframe>
It turned out the solution may be to replace www.youtube.com/embed/ with www.youtube-nocookie.com/embed/:
<iframe style="max-width: 100%" width="560" height="315"
src="https://www.youtube-nocookie.com/embed/FgxwCaL6UTA"
frameborder="0" allow="autoplay; encrypted-media" allowfullscreen="allowfullscreen"> </iframe>
(Weirdly on some refreshes of the page I get the same error. I'm not sure why that is.)