But the main use-case for this API seems to be organizing things - and it's strange that I would want certain tabs not take part in my organisation scheme.
If I had virtual desktop support, but certain windows are nevertheless shown on all desktops, that would feel broken, too.
Of course notifying users of audio/video recording is important, but this could be done with a separate notification - with options to stop recording or activate and un-hide the tab.
That's for a good reason. I don't want a tab I can't see to record audio or video. Why would you want to?