That’s an interesting idea. I’m trying to keep my themes as pure to the out-of-the-box WordPress experience as possible. Currently that means using featured image functionality. It would be great if WordPress let you set a video as featured, like with images. Then it would be easy to display them in the page and article templates.
What you are wanting to do is probably fairly difficult at this point. I can point you to a few things that might help you think through how best to tackle it. The theme is based on Bootstrap an it does support classes that indicate whether to display on desktop or mobile. See Bootstrap responsive utilities. You would have to embed both the video and the image and use those CSS classes to determine which to display.
WordPress also has a function wp_is_mobile() that can be used server-side (php). You’d have to edit the page.php and/or single.php template. See wp_is_mobile(). That function returns true for tablets though, not just for mobile phones.
Then there are additional complexities depending on whether you want text to overlay the video (like the theme does for images), whether you want the video to auto-play or not, etc.