Good news! There have been some updates, both at YouTube and at permies.com which makes this easier!
As an example, here is the full link from a YouTube music video that I copied:
https://www.youtube.com/watch?v=4W9VYY5QpLg
In the 'Reply' or 'New Topic' screen, there is a grey, Youtube (or Vimeo) button at the top (these are in the desktop view only) for making an embedded video:
--click the YouTube button
--a pop-up window asks for the URL
--paste in the URL (see my full link above as an example)
--click
OK
Easy-peasy! Here's what you get:
Now, on a mobile device, or if you click "Share" on YouTube, you get a shorter URL, in this case:
https://youtu.be/4W9VYY5QpLg
Again, using the permies YouTube button, it still works! (Thanks permies.com dev team for fixing this!!) See:
If you're in the mobile view of permies, and don't have the YouTube button, simply type:
"[youtube]"at the start of the YouTube URL
and
"[/youtube]" at the end of the YouTube URL
no spaces and no "(quotation)" marks of course, and note the
square brackets.
Hope that helps!