Controlling Netflix (HTML5) playback with Tampermonkey/javascript

I think this was a case of my thinking too hard about the problem. Once I stepped back, I realized that I had been looking in the right direction, but wasn't triggering the click event properly.

So, for example, to get the "button" that controls play and pause, you could use: document.getElementsByClassName("player-control-button player-play-pause")[0]. Then to click it programmatically in tampermonkey, you simply invoke the click event using:

document.getElementsByClassName("player-control-button player-play-pause")[0].click();

Volume and other controls in the bar are similar. The playback position is looking to be a bit trickier, but I'll do some more digging and add a comment to this answer once I figure it out.