There doesn’t seem to be any way to embed a video player on a thirdparty website. This makes it problematic to take advantage of the platform from within other content such as blogposts.
One feature I particularly care about is the ability to control the player from the embedding page. For easy interoperability with existing legacy code, I suggest following the youtube player api. The function names are sensible enough and offer all the functionality you could need.
There doesn’t seem to be any way to embed a video player on a thirdparty website.
Yes, there is: click on the « Share » button and you’ll have the URL of the video and the code for an iframe, allowing to embed the video on a website 
Nice. Silly me to miss such an obvious spot.
But it doesn’t address my second point, I’d like to control the embedded player from the embedding page.
could you open an issue on the project’s github issue tracker about that?
No problem, it’s # 458.