Sounds like an HDCP (DRM) issue.
There used to be a bug where Macs would generally behave weirdly with HDCP in multi-monitor scenarios (for MacBooks, this equates to a single external monitor). Not sure if that's still a thing but it might apply to you.
And unfortunately, Safari has always been flaky in this respect, so I'd recommend Chrome for sites serving DRM-protected content. (I say this as someone who generally prefers Safari)
You might have some luck by disconnecting HDMI cables, power-cycling the monitor(s) and Studio, then reconnecting cables. Make sure the devices actually turn off rather than going to sleep, and that cables are fully inserted.
For funsies, or to determine if DRM is in use somewhere: you can try taking a screenshot or screen-recording of a working playback session (e.g., in Chrome). Though you'll be able to see the video just fine on the display, it should appear pure black in any screenshot/recording.