> Obviously, faster load times should improve engagement.
I don't think it's this clear-cut. People can have different levels of expectation and tolerance for loading times of different things, and the page (post/article/whatever) is a different thing to the video embedded in it.
The load time of the rest of the page is also not necessarily limited by the loading time of the embed (depends what you're looking for). Arguably, the overall page load time will mask the content (embed) load time.
Finally, and perhaps most importantly, the loading time of the embed is a different thing to the loading time of the video. The metric to look at may be click-to-playback lag.