While I am not a fan of using Plugins and rather tend to implement everything myself, i’d like you to be aware of the fact, that the basic WordPress themes are always.. basic… and focus on all the core mechanics of the project itself.
If there is no feature request for featured videos yet, i recommend creating one and until its implementation (if ever) you should use a plugin because it is quite some work to do everything by your own if you want it to be usable in any way.
Even though it was last updated 4 years ago, you could give “Featured Video Plus” a try. According to the forums, it is still working. If you are a developer yourself, the code is open source so you can try extracting the core functionality.
cheers