This is a great use case for linking an image to a URL, because you can't embed a youtube video in an email. Even though the editor will allow you to do it, when the recipient receives the email, the video will not display/play.
Instead, as you have suggested, take a screen shot of a frame in the video (CNTRL-Shift-4/CMMD-Shift-4) from the youtube page. Then follow these instructions on how to add the image with a link back to the video:
button, and upload the image you previously saved. Now, when the image is clicked on, it will take the recipient to the link/url you saved.