Now that Manifest V3 has shipped and changed the available APIs for an extension from "Window" to "Worker" APIs, this is an appropriate topic for standardization work rather than implementation work: Should GetUserMedia be available on Worker?
So far, the response of the standards community has been a resounding "No" because all the concepts around prompting, displaying in-use indicators and so on have been strongly tied to the page context (tab) and its origin.
If you want this topic revisited, joining the W3C WEBRTC WG and arguing for it might be a good idea.