Thanks for the helpful pointers.
I had some initial success getting a basic video header working in the page header, using this tutorial for reference. Essentially I just added the relevant HTML to the page header template, and a tweaked version of the styles in my Less.
I would prefer to do something that was a bit more usable/configurable in the Authoring Tool - so it seems to me that a block extension like Ignacio's would be the way to go for this. (I guess I'd remove the page header from my overall theme and create a 'hero-image' type block that can be placed at the top of the page.) But not sure if this would introduce anything unexpected?
An important thing for my project would be to ensure this works on mobile. I gather this probably means the video should be replaced with an image on devices below a certain size.
The closest reference point I can find for how to do this in a Block element are in the couple of background switcher plugins that seem to be kicking around. Will give it a go and report back!
https://github.com/cgkineo/adapt-background-switcher/blob/master/less/background-switcher.less
https://github.com/mike-st/parallax-background-switcher/blob/master/less/background-switcher.less