Regretfully, the Content Area Modal does not have video integration. Please use the Lightbox Shortcode instead. The trade-off here though is that you'll use an image as a button. But, the result is, you have a responsive video lightbox.
Here's a sample. Insert this shortcode in a Content Area (not modal) element.
[image id="my-yt" src="https://picsum.photos/70/40" link="true" href="https://www.youtube.com/embed/b2Emnph8_L8" lightbox_video="true"]
Hope that helps.