Is there a solution so that the video can be hosted and played directly from my site and not streamed via Youtube?
The information on the following pages will help you.
https://www.wp101.com/10-reasons-why-you-should-never-host-your-own-videos/
As I mentioned, I have installed the background video plugin which gives me a shortcode, but I haven’t found a way to add a shortcode to my homepage.
You can achieve this by using custom plugin solution shared in the following topic and then adding the shortcode in the Content field option of [Shapely] Parralax Section For FrontPage widget.