What you're looking for is the functionality that is offered by plugin instead of theme. Here are some of the WordPress plugins suggestions that may suit your requirement.
1. WP Video Robot.
2. Video Import
3. YouTube Video Plugin
4. Cactus Bulk Video Theme Import
These are some of the options that you have that can work with any theme. And in your requirement for the video theme, I'd say go with simple pinterest clones and then use above plugins to embed the videos.