WordPress Media Video

My client has a video they don’t want to put on YouTube because it’s basically a “welcome to my website” video. I love FVE for embedding her YT stuff, but this one seems like a better candidate for uploading to the WP Media Library. Unfortunately it sort of stinks, layout-wise. Is it not possible to build this same sort of functionality into WP-hosted videos?

https://www.ads-software.com/plugins/fluid-video-embeds/