In another thread, users disabled hardware acceleration in Chrome or Firefox and were able to watch the video.
It appears the issue is with the way Apple processes displays in the hardware graphics pipeline vs. software. Somehow when the "Screen Recording" API is used, something with HDCP kicks in and prevents the video from displaying, but only when being processed in a particular way? It's a little confusing but we'll probably get a clean fix for this in, oh...a few years?